Cartan decomposition of a Lie group

Remarks

A Cartan involution exists and is unique up to inner automorphisms; this is the Cartan decomposition of the Lie algebra. The proof of 2 lets $G$ act on the symmetric space $G/K$, which has nonpositive curvature. Then the map $X \mapsto \exp(X) K$ is a diffeomorphism from $\mathfrak{p}$ onto $G/K$.

Example: $G = \mathrm{SL}_n(\R)$ with $\theta(X) = -X^{T}$. Then $\mathfrak{k} = \mathfrak{so}(n)$, $\mathfrak{p}$ is the space of symmetric matrices of trace $0$, and $K = \mathrm{SO}(n)$. The theorem is the polar decomposition: each $g \in \mathrm{SL}_n(\R)$ is uniquely $g = k e^{X}$ with $k$ orthogonal and $e^X$ symmetric positive definite.

Without a finite center, part 2 still holds with $K$ closed, but $K$ is no longer compact. For the universal cover of $\mathrm{SL}_2(\R)$, the subgroup $K$ is isomorphic to $\R$.

Going deeper

For a student

The theorem shows that $G$ is diffeomorphic to $K \times \R^{d}$ with $d = \dim \mathfrak{p}$. So the topology of $G$, for example its fundamental group, is the topology of the compact group $K$. The quotient $G/K$ is a Riemannian symmetric space of noncompact type, diffeomorphic to $\mathfrak{p}$. The Iwasawa decomposition $G = KAN$ and the decomposition $G = KAK$ refine this splitting.