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In mathematics, a complex Lie algebra is a Lie algebra over the complex numbers.
Given a complex Lie algebra g {\displaystyle {\mathfrak {g}}} , its conjugate g ¯ {\displaystyle {\overline {\mathfrak {g}}}} is a complex Lie algebra with the same underlying real vector space but with i = − 1 {\displaystyle i={\sqrt {-1}}} acting as − i {\displaystyle -i} instead. As a real Lie algebra, a complex Lie algebra g {\displaystyle {\mathfrak {g}}} is trivially isomorphic to its conjugate. A complex Lie algebra is isomorphic to its conjugate if and only if it admits a real form.