From we may define a positive subset by choosing the first set of indices from a basis of , and then the next set from a basis of . Since there are no real roots, the roots in have at least one nonvanishing component along , and the first nonzero one of these components is strictly positive. Since on , and since there are no real roots: . Thus permutes the simple roots, fixes the imaginary roots and exchanges in 2tuples the complex roots: it constitutes an involutive automorphism of the Dynkin diagram of .
A Vogan diagram is associated to the triple as follows. It corresponds to the standard Dynkin diagram of , with additional information: the 2element orbits under are exhibited by joining the correponding simple roots by a double arrow and the 1element orbit is painted in black (respectively, not painted), if the corresponding imaginary simple root is noncompact (respectively, compact).
The complex Lie algebra can be represented as the algebra of traceless complex matrices, the Lie bracket being the usual commutator. It has dimension 24. In principle, in order to compute the Killing form, one needs to handle the matrices of the adjoint representation. Fortunately, the uniqueness (up to an overall factor) of the biinvariant quadratic form on a simple Lie algebra leads to the useful relation
The coefficient appearing in this relation is known as the Coxeter index of .A Cartan–Weyl basis is obtained by taking the 20 nilpotent generators (with ) corresponding to matrices, all elements of which are zero except the one located at the intersection of row and column , which is equal to 1,
and the four diagonal ones, which constitute a Cartan subalgebra .The root space is easily described by introducing the five linear forms , acting on diagonal matrices as follows:
In terms of these, the dual space of the Cartan subalgebra may be identified with the subspace The 20 matrices are root vectors, i.e., is a root vector associated to the root .
By restricting ourselves to real combinations of these generators we obtain the real Lie algebra . The conjugation that it defines on is just the usual complex conjugation. This constitutes the split real form of . Applying the construction given in Equation (6.45) to the generators of , we obtain the set of antihermitian matrices
defining a basis of the real subalgebra . This is the compact real form of . The conjugation associated to this algebra (denoted by ) is minus the Hermitian conjugation, Since , induces a Cartan involution on , providing a Euclidean form on the previous subalgebra which can be extended to a Hermitian form on , Note that the generators and are real generators (although described by complex matrices) since, e.g., , i.e., .
The real forms of that are not isomorphic to or are isomorphic either to or . In terms of matrices these algebras can be represented as
We shall call these ways of describing the “natural” descriptions of . Introducing the diagonal matrix the conjugations defined by these subalgebras are given by:
The Dynkin diagram of is of type (see Figure 26).
Let us first consider an subalgebra. Diagonal matrices define a Cartan subalgebra whose all elements are compact. Accordingly all associated roots are imaginary. If we define the positive roots using the natural ordering , the simple roots , , are compact but is noncompact. The corresponding Vogan diagram is displayed in Figure 27.
However, if instead of the natural order we define positive roots by the rule , the simple positive roots are and which are compact, and and which are noncompact. The associated Vogan diagram is shown in Figure 28.
Alternatively, the choice of order leads to the diagram in Figure 29.
There remain seven other possibilities, all describing the same subalgebra . These are displayed in Figure 30.
In a similar way, we obtain four different Vogan diagrams for , displayed in Figure 31.
Finally we have two nonisomorphic Vogan diagrams associated with and . These are shown in Figure 32.
As we just saw, the same real Lie algebra may yield different Vogan diagrams only by changing the definition of positive roots. But fortunately, a theorem of Borel and de Siebenthal tells us that we may always choose the simple roots such that at most one of them is noncompact [129]. In other words, we may always assume that a Vogan diagram possesses at most one black dot.
Furthermore, assume that the automorphism associated with the Vogan diagram is the identity (no complex roots). Let be the basis of simple roots and its dual basis, i.e., . Then the single painted simple root may be chosen so that there is no with . This remark, which limits the possible simple root that can be painted, is particularly helpful when analyzing the real forms of the exceptional groups. For instance, from the Dynkin diagram of (see Figure 33), it is easy to compute the dual basis and the matrix of scalar products .
We obtain
from which we see that there exist, besides the compact real form, only two other nonisomorphic real forms of , described by the Vogan diagrams in Figure 34^{26}.Let us now illustrate the Cayley transformations. For this purpose, consider again with the imaginary diagonal matrices as Cartan subalgebra and the natural ordering of the defining the positive roots. As we have seen, is an imaginary noncompact root. The associated generators are
From the action of on the Cartan subalgebra , we may check that and that is such that and . Moreover commutes with . Thus constitutes a stable Cartan subalgebra with one noncompact dimension . Indeed, we have . If we compute the roots with respect to this new Cartan subalgebra, we obtain twelve complex roots (expressed in terms of their components in the basis dual to the one implicitly defined by Equations (6.97) and (6.98), six imaginary roots and a pair of real roots .Let us first consider the Cayley transformation obtained using, for instance, the real root . An associated root vector, belonging to , reads
The corresponding compact Cartan generator is which, together with the three generators in Equation (6.97), provide a compact Cartan subalgebra of .If we consider instead the imaginary roots, we find for instance that is a noncompact complex root vector corresponding to the root . It provides the noncompact generator which, together with
generates a maximally noncompact Cartan subalgebra of . A similar construction can be done using, for instance, the roots , but not with the roots as their corresponding root vectors and are fixed by and thus are compact.
We have seen that every real Lie algebra leads to a Vogan diagram. Conversely, every Vogan diagram defines a real Lie algebra. We shall sketch the reconstruction of the real Lie algebras from the Vogan diagrams here, referring the reader to [129] for more detailed information.
Given a Vogan diagram, the reconstruction of the associated real Lie algebra proceeds as follows. From the diagram, which is a Dynkin diagram with extra information, we may first construct the associated complex Lie algebra, select one of its Cartan subalgebras and build the corresponding root system. Then we may define a compact real subalgebra according to Equation (6.45).
We know the action of on the simple roots. This implies that the set of all roots is invariant under . This is proven inductively on the level of the roots, starting from the simple roots (level 1). Suppose we have proven that the image under of all the positive roots, up to level are in . If is a root of level , choose a simple root such that . Then the Weyl transformed root is also a positive root, but of smaller level. Since and are then known to be in , and since the involution acts as an isometry, is also in .
One can transfer by duality the action of on to the Cartan subalgebra , and then define its action on the root vectors associated to the simple roots according to the rules
These rules extend to an involution of .This involution is such that , with ^{27}. Furthermore it globally fixes , . Let and be the or eigenspaces of in . Define and so that . Set
Using , one then verifies that constitutes the desired real form of [129].
We shall exemplify the reconstruction of real algebras from Vogan diagrams by considering two examples of real forms of . The diagrams are shown in Figure 35.
The involutions they describe are (the upper signs correspond to the lefthand side diagram, the lower signs to the righthand side diagram):
Using the commutations relations we obtain Let us consider the lefthand side diagram. The corresponding eigenspace has the following realisation, and the eigenspace is given by The intersection then leads to the algebra and the remaining noncompact generator subspace becomesDoing the same exercise for the second diagram, we obtain the real algebra with , which is a 10parameter compact subalgebra, and given by
The following tables provide all real simple Lie algebras and the corresponding Vogan diagrams. The restrictions imposed on some of the Lie algebra parameters eliminate the consideration of isomorphic algebras. See [129] for the derivation.









Using these diagrams, the matrix defined by Equation (6.93), and the three matrices
we may check that the involutive automorphisms of the classical Lie algebras are all conjugate to one of the types listed in Table 25.

For completeness we remind the reader of the definitions of matrix algebras ( has been defined in Equation (6.93)):
Alternative definitions are:For small dimensions we have the following isomorphisms:
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