Complex Lie algebra

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

Real form

Given a complex Lie algebra ${\displaystyle \mathfrak{g}}$, a real Lie algebra ${\displaystyle {\mathfrak{g}}_0}$ is said to be a real form of ${\displaystyle \mathfrak{g}}$ if the complexification ${\displaystyle {\mathfrak{g}}_0{\otimes }_{\mathbb{R}}\mathbb{C}}$ is isomorphic to ${\displaystyle \mathfrak{g}}$.

A real form ${\displaystyle {\mathfrak{g}}_0}$ is abelian (resp. nilpotent, solvable, semisimple) if and only if ${\displaystyle \mathfrak{g}}$ is abelian (resp. nilpotent, solvable, semisimple). On the other hand, a real form ${\displaystyle {\mathfrak{g}}_0}$ is simple if and only if either ${\displaystyle \mathfrak{g}}$ is simple or ${\displaystyle \mathfrak{g}}$ is of the form ${\displaystyle \mathfrak{s}\times \overline{\mathfrak{s}}}$ where ${\displaystyle \mathfrak{s},\overline{\mathfrak{s}}}$ are simple and are the conjugates of each other.

The existence of a real form in a complex Lie algebra ${\displaystyle \mathfrak{g}}$ implies that ${\displaystyle \mathfrak{g}}$ is isomorphic to its conjugate; indeed, if ${\displaystyle \mathfrak{g}={\mathfrak{g}}_0{\otimes }_{\mathbb{R}}\mathbb{C}={\mathfrak{g}}_0\oplus i{\mathfrak{g}}_0}$, then let ${\displaystyle \tau :\mathfrak{g}\to \overline{\mathfrak{g}}}$ denote the ${\displaystyle \mathbb{R}}$-linear isomorphism induced by complex conjugate and then

${\displaystyle \tau (i(x+iy))=\tau (ix-y)=-ix-y=-i\tau (x+iy)}$,

which is to say ${\displaystyle \tau }$ is in fact a ${\displaystyle \mathbb{C}}$-linear isomorphism.

Conversely, suppose there is a ${\displaystyle \mathbb{C}}$-linear isomorphism ${\displaystyle \tau :\mathfrak{g}\stackrel{\sim }{\to }\overline{\mathfrak{g}}}$; without loss of generality, we can assume it is the identity function on the underlying real vector space. Then define ${\displaystyle {\mathfrak{g}}_0=\{z\in \mathfrak{g}|\tau (z)=z\}}$, which is clearly a real Lie algebra. Each element ${\displaystyle z}$ in ${\displaystyle \mathfrak{g}}$ can be written uniquely as ${\displaystyle z=2^{-1}(z+\tau (z))+i{2}^{-1}(i\tau (z)-iz)}$. Here, ${\displaystyle \tau (i\tau (z)-iz)=-iz+i\tau (z)}$ and similarly ${\displaystyle \tau }$ fixes ${\displaystyle z+\tau (z)}$. Hence, ${\displaystyle \mathfrak{g}={\mathfrak{g}}_0\oplus i{\mathfrak{g}}_0}$; i.e., ${\displaystyle {\mathfrak{g}}_0}$ is a real form.