6.1 Definitions
Lie algebras are usually, in a first step at least, considered as complex, i.e., as complex vector spaces,
structured by an antisymmetric internal bilinear product, the Lie bracket, obeying the Jacobi identity. If
denotes a basis of such a complex Lie algebra
of dimension
(over
), we
may also consider
as a real vector space of double dimension
(over
), a basis
being given by
. Conversely, if
is a real Lie algebra, by extending the field of
scalars from
to
, we obtain the complexification of
, denoted by
, defined as:
Note that
and
. When a complex Lie algebra
, considered
as a real algebra, has a decomposition
with
being a real Lie algebra, we say that
is a real form of the complex Lie algebra
. In other
words, a real form of a complex algebra exists if and only if we may choose a basis of the complex algebra
such that all the structure constants become real. Note that while
is a real space, multiplication by a
complex number is well defined on it since
. As we easily see from Equation (6.2),
where
and
.
The Killing form is defined by
The Killing forms on
and
or
are related as follows. If we split an arbitrary generator
of
according to Equation (6.2) as
, we may write:
Indeed, if
is a complex
matrix,
is a real
matrix: