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).^[2] 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.^[2]

The existence of a real form in a complex Lie algebra ${\displaystyle {\mathfrak {g}}}$  implies that ${\displaystyle {\mathfrak {g}}}$  is isomorphic to its conjugate;^[1] 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 ${\displaystyle \tau :{\mathfrak {g}}{\overset {\sim }{\to }}{\overline {\mathfrak {g}}}}$ ; 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 ${\displaystyle z=2^{-1}(z+\tau (z))+i2^{-1}(i\tau (z)-iz)}$ . 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.

Let ${\displaystyle {\mathfrak {g}}}$  be a semisimple complex Lie algebra that is the Lie algebra of a complex Lie group ${\displaystyle G}$ . Let ${\displaystyle {\mathfrak {h}}}$  be a Cartan subalgebra of ${\displaystyle {\mathfrak {g}}}$  and ${\displaystyle H}$  the Lie subgroup corresponding to ${\displaystyle {\mathfrak {h}}}$ ; the conjugates of ${\displaystyle H}$  are called Cartan subgroups.

Suppose there is the decomposition ${\displaystyle {\mathfrak {g}}={\mathfrak {n}}^{-}\oplus {\mathfrak {h}}\oplus {\mathfrak {n}}^{+}}$  given by a choice of positive roots. Then the exponential map defines an isomorphism from ${\displaystyle {\mathfrak {n}}^{+}}$  to a closed subgroup ${\displaystyle U\subset G}$ .^[3] The Lie subgroup ${\displaystyle B\subset G}$  corresponding to the Borel subalgebra ${\displaystyle {\mathfrak {b}}={\mathfrak {h}}\oplus {\mathfrak {n}}^{+}}$  is closed and is the semidirect product of ${\displaystyle H}$  and ${\displaystyle U}$ ;^[4] the conjugates of ${\displaystyle B}$  are called Borel subgroups.

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• Knapp, A. W. (2002). Lie groups beyond an introduction. Progress in Mathematics. Vol. 120 (2nd ed.). Boston·Basel·Berlin: Birkhäuser. ISBN 0-8176-4259-5..
• Serre, Jean-Pierre (2001). Complex Semisimple Lie Algebras. Berlin: Springer. ISBN 3-5406-7827-1.