Iwasawa decomposition

Remarks

The operators $\operatorname{ad} H$, $H \in \mathfrak{a}$, are symmetric for the inner product $B_\theta$ of the Cartan decomposition, and they commute, so they diagonalize together with real eigenvalues. So $\mathfrak{g}_0$ is the direct sum of the centralizer $\mathfrak{m} \oplus \mathfrak{a}$ of $\mathfrak{a}$, with $\mathfrak{m} \subseteq \mathfrak{k}$, and of the spaces $\mathfrak{g}_\lambda$, $\lambda \in \Sigma$. The Cartan involution $\theta$ maps $\mathfrak{g}_\lambda$ onto $\mathfrak{g}_{-\lambda}$. For $\lambda < 0$ and $X \in \mathfrak{g}_\lambda$, one writes $X = (X + \theta X) - \theta X$ with $X + \theta X \in \mathfrak{k}$ and $\theta X \in \mathfrak{n}$; this gives the sum in item 1, and a short argument gives directness. Example: for $\mathfrak{sl}_n(\R)$ with $\mathfrak{k} = \mathfrak{so}(n)$, one can take $\mathfrak{a}$ the diagonal matrices of trace $0$ and $\mathfrak{n}$ the strictly upper triangular matrices. Then item 1 is the Lie algebra form of the Gram–Schmidt process; the group form is the Iwasawa decomposition of a Lie group.

Going deeper

For a student

The Iwasawa decomposition refines the Cartan decomposition by splitting $\mathfrak{p}$ with the help of the restricted roots. It integrates to the decomposition $G = KAN$ of the group, which is basic in harmonic analysis on semisimple Lie groups and in the theory of parabolic subgroups.