### 6.7 The real semi-simple algebras

The dimensional reduction from 10 to 3 dimensions of supergravity coupled to Maxwell multiplets leads to a nonlinear sigma model with Lie= (see Section 7). To investigate the geometry of these cosets, we shall construct their Tits–Satake diagrams.

The Lie algebra can be represented by antisymmetric complex matrices. The compact real form is , naturally represented as the set of antisymmetric real matrices. One way to describe the real subalgebras , aligned with the compact form , is to consider as the set of infinitesimal rotations expressed in Pauli coordinates, i.e., to represent the hyperbolic space on which they act as a Euclidean space whose first coordinates, , are real while the last coordinates are purely imaginary. Writing the matrices of in block form as

where
we may obtain a maximal Abelian subspace by allowing to have nonzero elements only on its diagonal, i.e., to be of the form:
with or , respectively.

To proceed, let us denote by the matrices whose entries are everywhere vanishing except for a block,

on the diagonal. These matrices have the following realisation in terms of the (defined in Equation (6.83)):

They constitute a set of commuting generators that provide a Cartan subalgebra; it will be the Cartan subalgebra fixed by the Cartan involution defined by the real forms that we shall now discuss.

#### 6.7.1 Dimensions

Motivated by the dimensional reduction of supergravity, we shall assume , even. We first consider . Then by reordering the coordinates as follows,

we obtain a Cartan subalgebra of , with noncompact generators first, and aligned with the one introduced in Equation (6.177) by considering the basis . These generators are all orthogonal to each other. Let us denote the elements of the dual basis by , and split them into two subsets: and . The action of the Cartan involution on these generators is very simple,
The root system of is , represented by . A simple root basis can be taken as:
It is then straigthforward to obtain the action of the Cartan involution on the simple roots:
The corresponding Tits–Satake diagrams are displayed in Figure 38.

From Equation (6.179) we also obtain without effort that the set of restricted roots consists of the roots , each of multiplicity one, and the roots , each of multiplicity . These constitute a root system.

#### 6.7.2 Dimensions

Following the same procedure as for the previous case, we obtain a Cartan subalgebra consisting of noncompact generators and compact generators. The corresponding Tits–Satake diagrams are displayed in Figure 39.

The restricted root system is now of type , with long roots of multiplicity one and short roots of multiplicity .

#### 6.7.3 Dimensions

Here the root system is of type , represented by , where the orthonormal vectors again constitute a basis dual to the natural Cartan subalgebra of . Now, and are both assumed even, and we may always suppose . The Cartan involution to be considered acts as previously on the :

and
The simple roots can be chosen as

on which the Cartan involution has the following action:

• For
• For
• For

The corresponding Tits–Satake diagrams are obtained in the same way as before and are displayed in Figure 40.

When , the restricted root system is again of type , with long roots of multiplicity one and short roots of multiplicity . For , the short roots disappear and the restricted root system is of type, with all roots having multiplicity one.