6.2 A preliminary example: 
Before we proceed to develop the general theory of real forms, we shall in this section discuss in detail
some properties of the real forms of
. This is a nice example, which exhibits many properties
that turn out not to be specific just to the case at hand, but are, in fact, valid also in the general framework
of semi-simple Lie algebras. The main purpose of subsequent sections will then be to show how
to extend properties that are immediate in the case of
, to general semi-simple Lie
algebras.
6.2.1 Real forms of 
The complex Lie algebra
can be represented as the space of complex linear combinations of the
three matrices
which satisfy the well known commutation relations
A crucial property of these commutation relations is that the structure constants defined by the brackets
are all real. Thus by restricting the scalars in the linear combinations from the complex to the real
numbers, we still obtain closure for the Lie bracket on real combinations of
and
,
defining thereby a real form of the complex Lie algebra
: the real Lie algebra
.
As we have indicated above, this real form of
is called the “split real form”.
Another choice of
generators that, similarly, leads to a real Lie algebra consists in taking
times the Pauli matrices
,
,
, i.e.,
The real linear combinations of these matrices form the familiar
Lie algebra (a real Lie algebra,
even if some of the matrices using to represent it are complex). This real Lie algebra is non-isomorphic (as a
real algebra) to
as there is no real change of basis that maps
on a basis with the
commutation relations. Of course, the two algebras are isomorphic over the complex
numbers.
6.2.2 Cartan subalgebras
Let
be a subalgebra of
. We say that
is a Cartan subalgebra of
if it is a Cartan
subalgebra of
when the real numbers are replaced by the complex numbers. Two Cartan
subalgebras
and
of
are said to be equivalent (as Cartan subalgebras of
) if
there is an automorphism
of
such that
.
The subspace
constitutes clearly a Cartan subalgebra of
. The adjoint action of
is
diagonal in the basis
and can be represented by the matrix
Another Cartan subalgebra of
is given by
, whose adjoint action with respect to
the same basis is represented by the matrix
Contrary to the matrix representing
, in addition to 0 this matrix has two imaginary eigenvalues:
. Thus, there can be no automorphism
of
such that
,
since
has the same eigenvalues as
, implying that the eigenvalues of
are necessarily real
(
).
Consequently, even though they are equivalent over the complex numbers since there is an
automorphism in
that connects the complex Cartan subalgebras
and
, we obtain
The real Cartan subalgebras generated by
and
are non-isomorphic over the real numbers.
6.2.3 The Killing form
The Killing form of
reads explicitly
in the basis
. The Cartan subalgebra
is spacelike while the Cartan subalgebra
is
timelike. This is another way to see that these are inequivalent since the automorphisms of
preserve the Killing form. The group
of automorphisms of
is
, while the
subgroup
of inner automorphisms is the connected component
of
. All spacelike directions are equivalent, as are all timelike directions, which shows that
all the Cartan subalgebras of
can be obtained by acting on these two inequivalent
particular ones by
, i.e., the adjoint action of the group
. The lightlike
directions do not define Cartan subalgebras because the adjoint action of a lighlike vector is
non-diagonalizable. In particular
and
are not Cartan subalgebras even though they are
Abelian.
By exponentiation of the generators
and
, we obtain two subgroups, denoted
and
:
The subgroup defined by Equation (6.14) is noncompact, the one defined by Equation (6.15) is
compact; consequently the generator
is also said to be noncompact while
is called
compact.
6.2.4 The compact real form 
The Killing metric on the group
is negative definite. In the basis
, it reads
The corresponding group obtained by exponentiation is
, which is isomorphic to the 3-sphere
and which is accordingly compact. All directions in
are equivalent and hence, all Cartan subalgebras
are
conjugate to
. Any generator provides by exponentiation a group isomorphic to
and is thus compact.
Accordingly, while
admits both compact and noncompact Cartan subalgebras, the Cartan
subalgebras of
are all compact. The real algebra
is called the compact real form of
. One often denotes the real forms by their signature. Adopting Cartan’s notation
for
, one has
and
. We shall verify before that there are no other
real forms of
.
6.2.5
and
compared and contrasted – The Cartan involution
Within
, one may express the basis vectors of one of the real subalgebras
or
in
terms of those of the other. We obtain, using the notations
and
:
Let us remark that, in terms of the generators of
, the noncompact generators
and
of
are purely imaginary but the compact one
is real.
More precisely, if
denotes the
conjugation
of
that fixes
, we obtain:
or, more generally,
Conversely, if we denote by
the conjugation of
that fixes the previous
Cartan
subalgebra in
, we obtain the usual complex conjugation of the matrices:
The two conjugations
and
of
associated with the real subalgebras
and
of
commute with each other. Each of them, trivially, fixes pointwise the algebra
defining it and globally the other algebra, where it constitutes an involutive automorphism
(“involution”).
The Killing form is neither positive definite nor negative definite on
: The symmetric matrices
have positive norm squared, while the antisymmetric ones have negative norm squared. Thus, by changing
the relative sign of the contributions associated with symmetric and antisymmetric matrices, one can obtain
a bilinear form which is definite. Explicitly, the involution
of
defined by
has
the feature that
is positive definite. An involution of a real Lie algebra with that property is called a “Cartan involution”
(see Section 6.4.3 for the general definition).
The Cartan involution
is just the restriction to
of the conjugation
associated with the
compact real form
since for real matrices
. One says for that reason that the compact
real form
and the noncompact real form
are “aligned”.
Using the Cartan involution
, one can split
as the direct sum
where
is the subspace of antisymmetric matrices corresponding to the eigenvalue
of the Cartan
involution while
is the subspace of symmetric matrices corresponding to the eigenvalue
.
These are also eigenspaces of
and given explicitly by
and
. One has
i.e., the real form
is obtained from the compact form
by inserting an “
” in front of the
generators in
.
6.2.6 Concluding remarks
Let us close these preliminaries with some remarks.
- The conjugation
allows to define a Hermitian form on
:
- Any element of the group
can be written as a product of elements belonging to the
subgroups
,
and
(Iwasawa decomposition),
- Any element of
is conjugated via
to a multiple of
,
so, denoting by
the (maximal) noncompact Cartan subalgebra of
, we obtain
- Any element of
can be written as the product of an element of
and an element of
. Thus, as a consequence of the previous remark, we have
(Cartan).
- When the Cartan subalgebra of
is chosen to be
, the root vectors are
and
. We
obtain the compact element
, generating a non-equivalent Cartan subalgebra by taking the
combination
Similarly, the normalized root vectors associated with
are (up to a complex phase)
:
Note that both the real and imaginary components of
are noncompact. They allow to obtain
the noncompact Cartan generators
by taking the combinations