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A representation of a real non-compact semi-simple Lie algebra  $  \mathfrak g $(
cf. [[Lie algebra, semi-simple|Lie algebra, semi-simple]]) as a direct sum of vector spaces (*). If  $  \mathfrak g ^ {\mathbf C } $
denotes the complexification (complex envelope) of  $  \mathfrak g $(
cf. [[Complexification of a Lie algebra|Complexification of a Lie algebra]]), then there exists in  $  \mathfrak g ^ {\mathbf C } $
a real compact subalgebra  $  \mathfrak g  ^ {k} $
of the same dimension as  $  \mathfrak g $
such that the following decompositions into direct sums of vector spaces hold:

$$ \tag{* }
\mathfrak g  ^ {k}  = \ 
\mathfrak t + \mathfrak p ,\ \ 
\mathfrak g  =  \mathfrak t +
i \mathfrak p ,
$$

where  $  \mathfrak t $
is the subalgebra of invariant elements of some involutory automorphism (involution)  $  \phi $
of  $  \mathfrak g  ^ {k} $
and  $  \mathfrak p $
is the set of anti-invariant elements of  $  \phi $. 
The second formula is the Cartan decomposition of  $  \mathfrak g $(
see [[#References|[1]]]). The Cartan decomposition reduces the classification of real non-compact semi-simple Lie algebras to that of compact semi-simple Lie algebras and involutory automorphisms in them.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Helgason,   "Differential geometry and symmetric spaces" , Acad. Press  (1962)</TD></TR></table>