Cartan decomposition

A representation of a real non-compact semi-simple Lie algebra $ \mathfrak g $( cf. 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), 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 [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

[1]	S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press (1962)