A Cartan decomposition of a real semisimple Lie algebra 
is a vector
space decomposition 
into the 
and 
eigenspaces of a Cartan involution. It satisfies ![[k,k] subset= k](/images/equations/CartanDecomposition/Inline5.svg)
, ![[k,p] subset= p](/images/equations/CartanDecomposition/Inline6.svg)
, and ![[p,p] subset= k](/images/equations/CartanDecomposition/Inline7.svg)
.
Cartan Decomposition
See also
Lie Algebra, Semisimple Lie AlgebraExplore with Wolfram|Alpha
References
Helgason, S. Differential Geometry, Lie Groups, and Symmetric Spaces. Providence, RI: Amer. Math. Soc., 2001.Cite this as:
Weisstein, Eric W. "Cartan Decomposition." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CartanDecomposition.html