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
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
)):
Motivated by the dimensional reduction of supergravity, we shall assume , even. We first consider
. Then by reordering the coordinates as follows,
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.
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
.
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
:
on which the Cartan involution has the following action:
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.
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