I hope to know the classification of real forms of complex simple Lie algebras of types $A$, $D$, $E$ up to inner automorphisms.


Let $\mathfrak{g}\_1$ and $\mathfrak{g}\_2$ be real forms of a complex simple Lie algebra $\mathfrak{g}$.
We say that they are equivalent if there is an isomorphism $\mathfrak{g}\_1 \to \mathfrak{g}\_2$
which extends to an inner automorphism of $\mathfrak{g}$. 
However, there may be real forms which are isomorphic but not equivalent. 



> 
> Where can we find the classification of such equivalence classes of real forms?
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> 
> 



Classification of real forms up to inner automorphisms
The one case (for $\frak g$ simple) where the two definitions of real forms (up to $Aut(\frak g)$ versus $Int(\frak g)$) don't agree is the following.


There is a real form of $\frak g=\frak s\mathfrak o(2n,\mathbb C)$ denoted 
$\frak s\frak o^\*(2n)$. 


For $n\ge4$ even $\frak g$ has two subalgebras 
isomorphic to $\frak s\frak o^\*(2n)$ 
which are related by $Aut(\frak g)$, but not by $Int(\frak g)$.
If $n\ge 3$ is odd there is only one such algebra up to $Int(\frak g)$.
The difference is because the center of the simply connected group is $\mathbb Z/2\times\mathbb Z/2$ if $n$ is even, and $\mathbb Z/4$ if $n$ is odd.


$n=4$ is particularly interesting: the triality automorphism of $D\_4$ interchanges these 
two copies of $\mathfrak s\mathfrak o\*(2n)$, as well as $\frak s\frak o(6,2)\simeq \frak s\frak o^\*(2n)$.


