Real form (Lie theory)

In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers. A real Lie algebra g₀ is called a real form of a complex Lie algebra g if g is the complexification of g₀:

${\displaystyle {\mathfrak {g}}\simeq {\mathfrak {g}}_{0}\otimes _{\mathbb {R} }\mathbb {C} .}$

The notion of a real form can also be defined for complex Lie groups. Real forms of complex semisimple Lie groups and Lie algebras have been completely classified by Élie Cartan.