Lie algebra
Real form and complexification

Given a complex Lie algebra g {\displaystyle {\mathfrak {g}}}, a real Lie algebra g 0 {\displaystyle {\mathfrak {g}}_{0}} is said to be a real form of g {\displaystyle {\mathfrak {g}}} if the complexification g 0 ⊗ R C {\displaystyle {\mathfrak {g}}_{0}\otimes _{\mathbb {R} }\mathbb {C} } is isomorphic to g {\displaystyle {\mathfrak {g}}}. A real form need not be unique; for example, s l (2, C) {\displaystyle {\mathfrak {sl}}(2,\mathbb {C})} has two real forms up to isomorphism, s l (2, R) {\displaystyle {\mathfrak {sl}}(2,\mathbb {R})} and s u (2) {\displaystyle {\mathfrak {su}}(2)}.

Given a semisimple complex Lie algebra g {\displaystyle {\mathfrak {g}}}, a split form of it is a real form that splits; i.e., it has a Cartan subalgebra which acts via an adjoint representation with real eigenvalues. A split form exists and is unique (up to isomorphism). A compact form is a real form that is the Lie algebra of a compact Lie group. A compact form exists and is also unique up to isomorphism.
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