Cartan decomposition

The Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure theory and representation theory. It generalizes the polar decomposition or singular value decomposition of matrices. Its history can be traced to the 1880s work of Élie Cartan and Wilhelm Killing.

Let ${\displaystyle {\mathfrak {g}}}$ be a real semisimple Lie algebra and let ${\displaystyle B(\cdot ,\cdot )}$ be its Killing form. An involution on ${\displaystyle {\mathfrak {g}}}$ is a Lie algebra automorphism ${\displaystyle \theta }$ of ${\displaystyle {\mathfrak {g}}}$ whose square is equal to the identity. Such an involution is called a Cartan involution on ${\displaystyle {\mathfrak {g}}}$ if ${\displaystyle B_{\theta }(X,Y):=-B(X,\theta Y)}$ is a positive definite bilinear form.

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