Los Vasco Grande Reserve 2006 Cabernet Sauvignon

While the Cabernet Sauvignon 2006 is light and balanced with good persistent tannins, the Grand Reserve is full bodied and dry. It is big in style with powerful and juicy plum and dark cherry fruit flavours on the palate. Well balanced with good complexity and a long finish that has layers of spice and subtle hints of cocoa.

Best decanted for an hour to really open up the wine before drinking.

Hardys Oomoo Unwooded Chardonnay 2005

This is the inaugural vintage of Oomoo Unwooded Chardonnay produced at the historic Tintara Winery in McLaren Vale. It represents all the great qualities of a McLaren white with its reliability and abundance of ripe fruit flavours.

The Oomoo Unwooded offers a tight and full palate that carries pleasant, well balanced creamy Chardonnay tones set against a backdrop of tart but mellow citrussy flavours. Pale yellow with green hues, this wine displays lifted aromas of fresh peach, melon and fig with underlying citrus tones. There lies a perfect balance between sweet fruitiness and cakey, dry white qualities. The wine sits comfortably on the palate and is full flavoured. It shows characters of ripe peach, honeydew melon and lemon combined with a long elegant finish.

James Halliday’s 2005 Top 100 (selected)
Reds under $25
Hardys Oomoo Shiraz 2004 (94 points $12.95)
Hackersley Merlot 2004 (95 points $24)
Chalkers Crossing Hilltops Cabernet Sauvignon 2004 (95 points $24)
Ferngrove Majestic Cabernet Sauvignon 2003 (95 points $25)

Whites over $20
Peter Lehmann Reserve Riesling 2001 (96 points $24)
Hardys Eileen Hardy Chardonnay 2002 (96 points $38)
Leeuwin Estate Art Series Chardonnay 2002 (97 points $80)
Tyrrell’s Vat 1 Semillon 1999 (97 points $40)

Chateau Potensac 2004 Cru Bourgeois Exceptionnel

The vineyards of Potensac are located in Ordonnac, in the Médoc appellation, and incorporate the vines of three properties managed as a single entity, these being Potensac, Gallais-Bellevue and Lassalle. There is a rigorous selection for the grand vin Chateau Potensac, with about 40-45% of the crop going to the second wine, which today is bottled as La Chapelle de Potensac, although Chateau Lassalle has also been used as a second label in the past.

During recent decades the vineyard has been slightly dominated by Cabernet Sauvignon which accounts for about 60% of the vines, with approximately 25% Merlot and 15% Cabernet Franc in addition, planted at an average 8000 vines/ha. But with the purchase of new Merlot vines there is naturally a swing towards this variety, and it is notable that the 2005 vintage included more Merlot than Cabernet (41% Merlot, 40% Cabernet Sauvignon, 19% Cabernet Franc) in the final blend. Yields are restricted to approximately 35 hl/ha, and once harvested by hand the fruit is fermented at a maximum temperature of 28ºC in stainless steels and concrete vats, with 15 to 18 days maceration and constant pumping over.

Like Sociando-Mallet, Potensac is yet another chateau which illustrates the defunct nature of the 1855 classification of Bordeaux. Potensac regularly turns out wines of classed growth quality, but has only a Cru Bourgeois designation, although in the Cru Bourgeois classification of in 2003 (which subsequently collapsed following a legal challenge), it was accorded Cru Bourgeois Exceptionnel status, a short-lived recognition of the quality to be found here.

2004
Youthful hue, quite attractive, although this isn't matched on the nose which is quite closed down, with just a suggestion of some dark fruits when the wine is worked hard. A cool style, quite well textured, good tannic grip, but it is in keeping with the rest of the wine. An admirable style with good potential.

1996
A dark, claretty hue, with a cherry red rim. The nose is dark, smoky, with crisply bright yet deep, meaty fruit. More of the same on the palate, where the fruit has a precise, admirable presence with a warm, roasted yet fresh style. This has a very well defined structure, very upright, classic in nature, with a fine grip beneath. There is some bitterness to the fruit, which adds a delightful complexity.

Grape varietals : 46% Cabernet-sauvignon, 16% Cabernet-franc, 36% Merlot, 2% Carmenere.

Quotient Group - recall Weyl Group in SU(3)

Normal Subgroup and Equivalence

Let G be a group and N a subgroup of G. Then, N is called a normal subgroup of G if for each element g of G and each element n of N, the element gng^-1 belongs to N. Note that if G is commutative, then every subgroup of G is automatically normal because gng^-1 = n. 

If G and H are groups and Φ : G --> H is a homomorphism, then ker Φ is a normal subgroup of G. Let e₂ denote the identity element of H and suppose n is an element of ket Φ (i.e., Φ(n) = e₂). Then Φ(gng^-1) = e₂. Thus, ket Φ is a normal subgroup of G. 

Suppose G is a group and N is a normal subgroup of G. Define two elements g and h to be equivalent if gh^-1 ∈ N.

1. If g₁ is equivalent to g₂ and h₁ is equivalent to h₂, then g₁h₁ is equivalent to g₂h₂.

2. If g₁ is equivalent to g₂, then g₁-1 is equivalent to g₂-1 .

Quotient Group

If g is any element of G, let [g] denote the equivalence class containing G; that is [g] is the subset of G consisting of all elements equivalent to g (including g itself). 

Let G be a group and N a normal subgroup of G. The quotient group G/N is the set of all equivalence classes in G, with the product operation defined by [g][h] = [gh]. The elements of G/N are equivalence classes. The group product is defined by choosing one element g out of the first equivalence class, choosing one element h out of the second equivalence class, and then define the product to be the equivalence class containing gh. 

The idea behind the quotient group construction is that we make a new group out of G by setting every element of N equal to the identity. This then forces ng to be equal to g ∈ G and n ∈ N. However, ng and n are equivalent, since (ng)g^-1 = n ∈ N. So setting elements of N equal to the identity forces elements that are equivalent to be equal. The condition that N be a normal subgroup guarantees that we still have defined group operations after setting equivalent elements equal to each other. 

If G is a group and N is a normal subgroup, then there is a homomorphism q of G into the quotient group G/N given by q(g) = [g]. Follows from the definition of the product operation on G/N that q is indeed a homomorphism and it clearly maps G onto G/N. For Φ : G --> H is homomorphism, we have seen that ker Φ is a normal subgroup of G. If Φ maps G onto H, then it can be shown that H is homomorphic to the quotient group G/ker Φ. 

Note: If G is a matrix Lie group, then G/N may not be. Even if G/N happens to be a matrix Lie group, there is no canonical procedure for finding a matrix representation of it. 

Examples

1. Group of integers modulo n. In this case, G = Z and N = nZ (set of integer multiples of n). To form the quotient group, we say that two elements of Z are equivalent if their difference is in N. Thus, the equivalence class of an integer i is the set of all integers that are equal modulo n to i.

2. Taking G = SL(n;C) and taking N to be the set of elements of SL(n;C) that are multiples of the identity. The elements of N are the matrices of the form e^2πik/nI, k = 0, 1, ...., n-1. This is a normal subgroup of SL(n;C) because each element of N is a multiple of the identity and, thus, for any A ∈ SL(n;C), we have A(e^2πik/nI)A^-1 = AA^-1(e^2πik/nI) = e^2πik/nI. The quotient group SL(n;C)/N is customarily denoted PSL(n;C), where P stands for "projective". It can shown that PSL(n;C) is a simple group for all n ≥ 2; that is, PSL(n;C) has no normal subgroups other than {I} and PSL(n;C) itself. 

Abstract Root Systems

Root System

A root system is a finite-dimensional real vector space E with an inner product <⋅,⋅>, together with a finite collection R of nonzero vectors in E satisfying the following properties:

1. The vectors in R span E.

2. If α is in R, then so is -α.

3. If α is in R, then the only multiples of α in R are α and -α.

4. If α and β are in R, then so is w_α⋅β, where w_α is the linear transformation of E defined by w_α⋅β = β - 2(<β,α>/<α,α>)α,  β ∈ E. Note: w_α⋅α = -α.

5. For all α and β in R, the quantity 2<β,α>/<α,α> is an integer.

The map w_α is the reflection about the hyperplane perpendicular to α; that is, w_α⋅α = -α and w_α⋅β = β for all β in E that are perpendicular to α. It should be evident that  w_α is an orthogonal transformation of E with determinant -1. 

Since the orthogonal projection of β onto α is given by (<β,α>/<α,α>)α, we note that the quantity 2<β,α>/<α,α> is twice that the coefficient of α in this projection. Thus, the it is equivalent to saying that the projection of β onto α is an integer or half-integer multiple of α. 

Suppose (E,R) and (F,S) are root systems. Consider the vector space E ⊕ F, with the natural inner product determined by the inner products on E and F. Then, R ∪ S is a root system in E ⊕ F, called the direct sum of R and S. 

A root system (E,R) is called reducible if there exists an orthogonal decomposition E = E₁⊕ E₂ with dim E₁ > 0 and dim E₂ > 0 such that every element in R is either in E₁ or in E₂. If no such decomposition exists, (E,R) is called irreducible.

Two root systems (E,R) and (F,S) are said to be equivalent if there exists an invertible linear transformation A : E --> F such that A maps R onto S and such that for all α ∈ R and β ∈ E, we have A(w_α⋅β ) = w_α⋅Aβ. A map A with this property is called an equivalence. Note that the linear map A is not required to preserve inner products, but only to preserve the reflections about the roots. 

Rank

The dimension of E is called the rank of the root system and the elements of R are called roots.

The A₁ rank-one root system R must consist of a pair {α, -α}, where α is a nonzero element of E. 

In rank-two root system, there are four possibilities: A₁ x A₁, A₂, B₂, and G₂. In A₁ x A₁, the lengths of the horizontal roots are unrelated to the lengths of the vertical roots. In A₂, all roots have the same length; angle between successive roots is 60º. In B₂, the length of the longer roots is √2 times the length of the shorter roots;  angle between successive roots is 45º. In G₂, the length of the longer roots is √3 times the length of the shorter roots;  angle between successive roots is 30º. 

Weyl Group

If (E, R) is a root system, then the Weyl group W of R is the subgroup of the orthogonal group of E generated by the reflections w_α, α ∈ R. By assumption, each w_α maps R into itself, indeed onto itself, since each α ∈ R satisfies α = w_α⋅ (w_α⋅α). It follows that every element of W maps R onto itself.

Since the roots span E, a linear transformation of E is determined by its action on R.  This shows that the Weyl group is a finite subgroup of O(E) and may be regarded as a subgroup of the permutation group on R.

Semisimple Lie Group

There are three equivalent characterization of semisimple Lie algebras. The first characterization is one which is isomorphic to a direct sum of simple Lie algebras. The second characterization is that complexification of the Lie algebra of a compact simply-connected group, for example, that sl(n;C) ≅ su(n)_C is semisimple. The third characterization is that a Lie algebra g is semisimple if and only if it has the complete reducibility property, that is, if and only if every finite-dimensional representation of g decomposes as a direct sum of irreducibles. 

Recall that a group or Lie algebra is said to have the complete reducibility property if every finite-dimensional representation of it decomposes as a direct sum of irreducible invariant subspaces. A connected compact matrix Lie group always has this property. It follows that the Lie algebra of compact simply-connected matrix Lie group also has the complete reducibility property, since there is a one-to-one correspondence between the representations of the compact group and its Lie algebra. Because there is a one-to-one correspondence between the representations of a real Lie algebra and the complex-linear representations of its complexification, we see also that if a complex Lie algebra g is isomorphic to the complexification of the Lie algebra of a compact simply-connected group, then g has the complete reducibility property. We have seen this reasoning to sl(2;C) (the complexification of the Lie algebra of SU(2) and to sl(3;C) and the complexification of the Lie algebra SU(3)).

Complex semisimple Lie algebras are complex Lie algebras that are isomorphic to the complexification of the Lie algebra of a compact simply-connected matrix Lie group.

Definition

If g is a complex Lie algebra, the an ideal in g is a complex subalgebra h of g with the property that for all X in g and H in h, we have [X, H] in h. 

A complex Lie algebra g is called indecomposable if the only ideals in g are g and {0}. A complex Lie algebra g is called simple if g is indecomposable and dim ≥ 2.

A complex Lie algebra is called reductive if it is isomorphic to a direct sum of indecomposable Lie algebras. A complex Lie algebra is called semisimple if it is isomorphic to a direct sum of simple Lie algebras. Note that a reductive Lie algebra is a direct sum of indecomposable algebras, which are either simple or one-dimensional commutative. Thus, a reductive Lie algebra is one that decomposes as a direct sum of a semisimple algebra and a commutative algebra. 

The following table lists the complex Lie algebras that are either reductive (not semisimple) or semisimple.

sl(n;C) (n≥2) semisimple
so(n;C) (n≥3) semisimple 
so(2;C) reductive 
gl(n;C) (n≥1) reductive 
sp(n;C) (n≥1) semisimple

All of the above listed semisimple algebras are actually simple, except for so(4;C), which is isomorphic to sl(2;C) ⊕ sl(2;C). Every complex simple Lie algebra is isomorphic to one of sl(n;C), so(n;C) (n≠4), sp(n;C), or to one of the five "exceptional" Lie algebras conventionally called G₂, F₄, E₆, F₇, and E₈.

For real Lie algebra,

su(n) (n≥2) semisimple 
so(n) (n≥3)  semisimple 
so(2) reductive 
sp(n) (n≥1) semisimple 
so(n,k) (n+k ≥3) semisimple 
so(1,1) reductive 
sp(n;R) (n≥1) semisimple 
sl(n;R) (n≥2) semisimple
gl(n;R) (n≥1) reductive

In each case, the complexification of the listed Lie algebra is isomorphic to one of the complex Lie algebras in the above table. Note that the Heisenberg group, the Euclidean group, and the Poincare group are neither reductive nor semisimple. 

Highest Weight

If we have a representation with a weight μ = (m₁, m₂), then by applying the root vectors X_α X₂, X₃, Y₁, Y₂, Y₃, we can get some new weights of the form μ + α, where α is the root (recall that π(H₁)π(Z_α)v = (m₁+ a₁)π(Z_α)v). If π(Z_α)v = 0, then μ + α is not necessarily a weight. In analogy to the classification of the representation sl(2;C) : In each irreducible representation of sl(2;C), π(H) is diagonalizable, and there is a largest eigenvalue of π(H). Two irreducible representations of sl(2;C) with the same largest eigenvalue are equivalent. The highest eigenvalue is always a non-negative integer, and, conversely, for every non-negative integer m, there is an irreducible representation with highest eigenvalue m. 

Let α₁ = (2, -1) and  α₂ = (-1, 2) be the roots introduced in "Weights & Roots". Let μ₁  and μ₂ be two weights. Then, μ₁is higher than μ₂ if μ₁- μ₂ can be written in the form  μ₁ - μ₂ = aα₁ + bα₂ with a ≥ 0 and b ≥ 0. If π is a representation of sl(3;C), then a weight μ₀ for π is said to be a highest weight if for all weights μ of π, μ ≤ μ₀.

Note that the relation of "higher" is only a partial ordering because μ₁ is neither higher nor lower than μ₂. For example, {0, α₁ - α₂} has no highest element. Moreover, the coefficients a and b do not have to be integers, even if both μ₁ and μ₂ have integer entries. For example, (1,0) is higher than (0,0) since (1,0) = 2/3α₁ + 1/3α₂.

Theorem of Highest Weight

The theorem of highest weight is a main theorem regarding the irreducible representation of sl(3;C).

1. Every irreducible representation π of sl(3;C) is the direct sum of its weight spaces; that is π(H₁) and  π(H₂) are simultaneously diagonalizable in every irreducible representation.

2. Every irreducible representation of sl(3;C) has a unique highest weight μ₀, and two equivalent irreducible representations have the same highest weight.

3. two irreducible representations of sl(3;C) with the same highest weight are equivalent.

4. If π is an irreducible representation of sl(3;C), then the highest weight μ₀ of π is of the form μ₀ = (m₁, m₂) with m₁ and m₂ being non-negative integers. 

An ordered pair (m₁, m₂) with m₁ and m₂ being non-negative integers is called a dominant integral element. The theorem says that the highest weight of each irreducible representation of sl(3;C) is a dominant integral element and, conversely, that every dominant integral element occurs as the highest weight of some irreducible representation. 

However, if μ has integer coefficients and is higher than zero, this does not necessarily mean that μ is dominant integral. For example, α₁ = (2, -1) is higher than zero but is not dominant integral. Note that the condition on which weights can be highest weights is m₁ and m₂ being non-negative integer.

The dimension of the irreducible representation with highest weight (m₁, m₂) is

1/2(m₁ + 1)(m₂ + 1)(m₁ + m₂ + 2).

Weyl Group in SU(3)

The representations of sl(3;C) are invariant under the adjoint action of SU(3). Let π be a finite-dimensional representation of sl(3;C) acting on a vector space V and let Π be the associated representation of SU(3) acting on the same space. For any A ∈ SU(3), we can define a new representation π_A of sl(3;C), acting on the same vector space V, by setting π_A(X) = π(AXA^-1). Since the adjoint action of A on sl(3;C) is a Lie algebra automorphism, this is a representation of sl(3;C). Π(A) is an intertwining map between (π, V) and (π_A, V). We say that adjoint action of SU(3) is a symmetry of the set of equivalence classes of representations of sl(3;C). 

The two-dimensional subspace h of sl(3;C) spanned by H₁ and H₂ is called a Cartan subalgebra. In general, the adjoint action of A  ∈ SU(3) will not preserve the space h and so the equivalence of π and π_A does not tell us anything about the weights of π. However, there are elements A in SU(3) for which Ad_A does preserve h. These elements make up the Weyl group for SU(3) and give rise to a symmetry of the set of weights of any representation π. 

Let N be the subgroup of SU(3) consisting of those A ∈ SU(3) such that Ad_A(H) is an element of h for all H in h. And let Z be the subgroup of SU(3) consisting of those  A  ∈ SU(3) such that Ad_A(H) = H for all H ∈ h. The Weyl group of SU(3), denoted by W, is the quotient group N/Z. 

The group Z consists precisely of the diagonal matrices inside SU(3), namely the matrices of the form A = (e^iθ,0,0 : 0,e^iΦ,0 : 0,0,e^-i(θ+Φ)) for θ and Φ in R. The group N consists of precisely those matrices A ∈ SU(3) such that for each k = 1,2,3, there exist l ∈ {1,2,3} and θ ∈ R such that Ae_k = e^iθe_l. Here e₁, e₂,e₃ is the standard basis for C³. The Weyl group W = N/Z is isomorphic to the permutation group on three elements. 

In order to show that Weyl group is a symmetry of the weights of any finite-dimensional representation of sl(3;C), we need to adopt a less basis-dependent view of weights. A vector v is an eigenvector of π(H₁) and π(H₂) then it is also an eigenvector for π(H) for any element H of the space h spanned by H₁ and H₂. Furthermore, the eigenvalues must depend linearly on H. For π(J) = λ₂v, then π(aH + bJ)v = (aπ(H) + bπ(J))v = (aλ₁ + bλ₂)v. We have the following basis-independent notion of weight :  a linear functional μ ∈ h^* is called a weight for π if there exists a nonzero vector v in V such that π(H)v = μ(H)v for all H in h. Such a vector v is called a weight vector with weight μ. So a weight is just a collection of simultaneously eigenvalues of all the elements H of h, which depends linearly on H and, therefore, define a linear functional on h. The reason for adopting this basis-independent approach is that the action of the Weyl group does not preserve the basis {H₁, H₂} for h.

In another words, the Weyl group is a group of linear transformation of h. This means that W acts on h, and we denote this action as w⋅H. We can define an associated action on the dual space h^*. Thus, for μ ∈ h^* and w ∈ W, we define w⋅μ to be the element of  h^* given by (w⋅μ)(H) = μ(w^-1⋅H). 

Weights & Roots

There is a one-to-one correspondence between the finite-dimensional complex representations Π of SU(3) and the finite-dimensional complex-linear representation π of sl(3;C). This correspondence is determined by the property that Π(e^X) = e^π(X) for all X ∈ su(3) ⊂ sl(3;C). The representation Π is irreducible if and only if the representation π is irreducible. 

Simultaneous Diagonalization

Suppose that V is a vector space and A is some collection of linear operators on V. Then a simultaneous eigenvector for A is a nonzero vector v ∈ V such that for all A ∈ A, there exists a constant λ_A with Av = λ_Av. The numbers λ_A are the simultaneous eigenvalues associated to v. For example, consider the space D of all diagonal nxn matrices. For each k = 1,...,n, the standard basis element e_k is a simultaneous eigenvector for D. For each diagonal matrix A, the simultaneous eigenvalue associated to e_k is the k-th diagonal entry of A.

If A is a simultaneously diagonalisable family of linear operators on a finite-dimensional vector space V, then the elements of A commute.

If A is commuting collection of linear operators on a finite-dimensional vector space V and each A ∈ A is diagonalizable, then the elements of A are simultaneously diagonalizable. 

Basis for sl(3;C)

Every finite-dimensional represetation of sl(2;C) or sl(3;C) decomposes as a direct sum of irreducible invariant subspaces. Consider the following basis for sl(3;C):

H₁ = (1,0,0 : 0,-1,0 : 0,0,0), H₁ = (0,0,0 : 0,1,0 : 0,0,-1),

X₁ = (0,0,0 : 1,0,0 : 0,0,0), X₂ = (0,0,0 : 0,0,0 : 0,1,0), X₃ = (0,0,0 : 0,0,0 : 1,0,0),

Y₁ = (0,1,0 : 0,0,0 : 0,0,0), Y₂ = (0,0,0 : 0,0,1 : 0,0,0), Y₃ = (0,0,1 : 0,0,0 : 0,0,0).

The span of {H₁, X₁, Y₁} is a subalgebra of sl(3;C) which is isomorphic to sl(2;C) (can be seen by ignoring the third row and the third column in each matrix). Similarly for  {H₂, X₂, Y₂}. Thus, the following commutation relations exists:

[X₁, Y₁] = H₁, [X₂, Y₂] = H₂,

[H₁, X₁] = 2X₁, [H₂, X₂] = 2X₂,

[H₁, Y₁] = -2Y₁, [H₂, Y₂] = -2Y₂.

Other commute relations among the basis elements which involve at least one H₁ and H₂:

[H₁, H₂] = 0;

[H₁, X₁] = 2X₁, [H₁, Y₁] = -2Y₁,

[H₂, X₁] = -X₁, [H₂, Y₁] = Y₁;

[H₁, X₂] = -X₂, [H₁, Y₂] = Y₂,

[H₂, X₂] = 2X₂, [H₂, Y₂] = -2Y₂;

[H₁, X₃] = X₃, [H₁, Y₃] = -Y₃,

[H₂, X₃] = X₃, [H₂, Y₃] = -Y₃;

Adding all of the remaining commutation relations:

[X₁, Y₁] = H₁,

[X₂, Y₂] = H₂,

[X₃, Y₃] = H₁ + H₂;

[X₁, X₂] = X₃, [Y₁, Y₂] = -Y₃,

[X₁, Y₂] = 0, [X₂, Y₁] = 0;

[X₁, X₃] = 0, [Y₁, Y₃] = 0,

[X₂, X₃] = 0, [Y₂, Y₃] = 0;

[X₂, Y₃] = Y₁, [X₃, Y₂] = X₁,

[X₁, Y₃] = -Y₂, [X₃, Y₁] = -X₂.

Weights of sl(3;C)

A strategy to classify the representation sl(3;C) is to simultaneously diagonalize π(H₁) and π(H₂). Since H₁ and H₂ commute, π(H₁) and π(H₂) will also commute and so there is at least a chance that π(H₁) and π(H₂) can be simultaneously diagonalized. 

If (π, V) is a representation of sl(3;C), then an ordered pair μ = (m₁, m₂) ∈ C² is called a weight of π if there exists v ≠ 0 in V such that π(H₁)v = m₁v, π(H₂)v = m₂v. A nonzero vector v satisfying this is called a weight vector corresponding to the weight μ. If μ = (m₁, m₂) is a weight, then the space of all vectors v satisfying π(H₁)v = m₁v, π(H₂)v = m₂v is the weight space corresponding to the weight μ. The multiplicity of a weight is the dimension of the corresponding weight space. Equivalent representations have the same weights and multiplicities. 

If π is a representation of sl(3;C), then all of the weights of π are of the form μ = (m₁, m₂) with m₁ and m₂ being integers.

Roots of sl(3;C)

An ordered pair α = (a₁, a₂) ∈ C² is called a root if 

1. a₁ and a₂ are not both zero, and

2. there exists a nonzero Z ∈ sl(3;C) such that [H₁, Z] = a₁Z, [H₂, Z] = a₁Z. The element Z is called a root vector corresponding to the root α. 

Recall that ad_X(Y) = [X, Y], e^adx = Ad(e^X), and the adjoint mapping Ad_A(X) = AXA^-1. Condition 2 of above says that Z is a simultaneous eigenvector for ad_H1 and ad_H2.  This means that Z is a weight vector for the adjoint representation and weight (a₁, a₂). By condition 1 the roots are precisely the nonzero weights of the adjoint representation. 

There are six roots of sl(3;C). They form a "root system", called A₂.

α            Z

(2, -1)    X₁

(-1, 2)    X₂

(1, 1)     X₃

(-2, 1)    Y₁

(1, -2)    Y₂

(-1, -1)   Y₃

It is convenient to single out the two roots corresponding to X₁ and X₂ and given them special names: α₁ = (2, -1), α₂ = (-1,2). α₁ and α₂ are called the positive simple roots. They have the property that all of the roots can be expressed as linear combinations of α₁ and α₂ with integer coefficients :

(2, -1) = α₁

(-1, 2) = α₂

(1, 1) = α₁ + α₂

(-2, 1) = -α₁

(1, -2) = -α₂

(-1, -1) = -α₁ - α₂ .

Let α = (a₁, a₂) be a root and Z_α a corresponding root vector in sl(3;C). Let π be the representation of sl(3;C), μ = (m₁, m₂) a weight for π, and v ≠ 0 a corresponding weight vector. Then

π(H₁)π(Z_α)v = (m₁+ a₁)π(Z_α)v,

π(H₂)π(Z_α)v = (m₂+ a₂)π(Z_α)v.

Thus, either π(Z_α)v = 0 or π(Z_α)v is a new weight vector with weight

μ + α = (m₁+ a_1, m₂+ a₂) .