Complex Lie algebra
In mathematics, a complex Lie algebra is a Lie algebra over the complex numbers.

Given a complex Lie algebra , its conjugate  is a complex Lie algebra with the same underlying real vector space but with  acting as  instead. As a real Lie algebra, a complex Lie algebra  is trivially isomorphic to its conjugate. A complex Lie algebra is isomorphic to its conjugate if and only if it admits a real form (and is said to be defined over the real numbers).

Real form 

Given a complex Lie algebra , a real Lie algebra  is said to be a real form of  if the complexification  is isomorphic to .

A real form  is abelian (resp. nilpotent, solvable, semisimple) if and only if  is abelian (resp. nilpotent, solvable, semisimple). On the other hand, a real form  is simple if and only if either  is simple or  is of the form  where  are simple and are the conjugates of each other.

The existence of a real form in a complex Lie algebra  implies that  is isomorphic to its conjugate; indeed, if , then let  denote the -linear isomorphism induced by complex conjugate and then
,
which is to say  is in fact a -linear isomorphism.

Conversely, suppose there is a -linear isomorphism ; without loss of generality, we can assume it is the identity function on the underlying real vector space. Then define , which is clearly a real Lie algebra. Each element  in  can be written uniquely as . Here,  and similarly  fixes . Hence, ; i.e.,  is a real form.

Complex Lie algebra of a complex Lie group 
Let  be a semisimple complex Lie algebra that is the Lie algebra of a complex Lie group . Let  be a Cartan subalgebra of  and  the Lie subgroup corresponding to ; the conjugates of  are called Cartan subgroups.

Suppose there is the decomposition  given by a choice of positive roots. Then the exponential map defines an isomorphism from  to a closed subgroup . The Lie subgroup  corresponding to the Borel subalgebra  is closed and is the semidirect product of  and ; the conjugates of  are called Borel subgroups.