Complex Lie algebra

In mathematics, a complex Lie algebra is a Lie algebra over the complex numbers.
Given a complex Lie algebra formula_1, its conjugate formula_2 is a complex Lie algebra with the same underlying real vector space but with formula_3 acting as formula_4 instead. As a real Lie algebra, a complex Lie algebra formula_1 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 formula_1, a real Lie algebra formula_7 is said to be a real form of formula_1 if the complexification formula_9 is isomorphic to formula_1.
A real form formula_7 is abelian (resp. nilpotent, solvable, semisimple) if and only if formula_1 is abelian (resp. nilpotent, solvable, semisimple). On the other hand, a real form formula_7 is simple if and only if either formula_1 is simple or formula_1 is of the form formula_16 where formula_17 are simple and are the conjugates of each other.
The existence of a real form in a complex Lie algebra formula_18 implies that formula_18 is isomorphic to its conjugate; indeed, if formula_20, then let formula_21 denote the formula_22-linear isomorphism induced by complex conjugate and then
which is to say formula_24 is in fact a formula_25-linear isomorphism.
Conversely, suppose there is a formula_25-linear isomorphism formula_27; without loss of generality, we can assume it is the identity function on the underlying real vector space. Then define formula_28, which is clearly a real Lie algebra. Each element formula_29 in formula_1 can be written uniquely as formula_31. Here, formula_32 and similarly formula_24 fixes formula_34. Hence, formula_35; i.e., formula_7 is a real form.
Complex Lie algebra of a complex Lie group.
Let formula_1 be a semisimple complex Lie algebra that is the Lie algebra of a complex Lie group formula_38. Let formula_39 be a Cartan subalgebra of formula_1 and formula_41 the Lie subgroup corresponding to formula_39; the conjugates of formula_41 are called Cartan subgroups.
Suppose there is the decomposition formula_44 given by a choice of positive roots. Then the exponential map defines an isomorphism from formula_45 to a closed subgroup formula_46. The Lie subgroup formula_47 corresponding to the Borel subalgebra formula_48 is closed and is the semidirect product of formula_41 and formula_50; the conjugates of formula_51 are called Borel subgroups.