### 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: