E.B.Vinberg and A.L.Onishchik in their book give the following two definitions.

  1. For a complex Lie group $G$ its real Lie subgroup $H$ is called a real form of $G$, if

    a) the Lie algebra $L(H)$ of $H$ is a real form of the Lie algebra $L(G)$ of $G$, and

    b) $H$ has nonzero intersection with each connected component of $G$

  2. A (continuous) homomorphism $S:G\to G$ (of a complex Lie group $G$) is called a real structure on $G$, if

    a) $S(S(g))=g$, for $g\in G$, and

    b) the corresponding mapping of the Lie algebra $d S:L(G)\to L(G)$ is a real structure (another term: an involution) on $L(G)$.

It is known that on a connected complex Lie group (or on an irreducible algebraic group) $G$ for each real structure $S:G\to G$ the subgroup of stable elements $$ H=\{g\in G:\ Sg=g\} $$ is a real form on $G$, and the Lie algebra $L(H)$ coincides with the set of stable elements for $dS$: $$ L(H)=\{x\in L(G):\ dS(x)=x\}. $$

My question:

Is this a one-to-one correspondence between real forms and real structures (for a reasonable class of groups)?

In particular,

if $H$ is a real form of a complex a Lie group $G$, does there always exist a real structure $S:G\to G$ such that $H$ is the subgroup of stable elements for $S$ in $G$, and $L(H)$ is the subspace of stable elements for $dS$ in $L(G)$?

This is strange, I can't find a reference, even for the case, when

$H$ is a compact real Lie group, and $G$ is its complexification.

(Actually, that would be enough for me.)

I would be grateful for anybody's help.


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