Dimension of the maximal abelian subalgebra

[mpham26uchicago]

Let $\mathfrak{g} = \mathfrak{k} \otimes \mathfrak{m}$ be Cartan decomposition of the semi-simple algebra $\mathfrak{g}$. The following feels true...

$$\dim(\mathfrak{m}) - \dim(\mathfrak{h}) \leq \dim(\mathfrak{k})$$

Intuition. So $\mathfrak{m}$ can be expressed as parameters of $\mathfrak{h}$ and parameters of $\mathfrak{k}$. Assume that the number of parameters is equal to the dimension of the space, then it is impossible that

$$\dim(\mathfrak{k}) + \dim(\mathfrak{h}) > \dim(\mathfrak{m}).$$

[mpham26uchicago]

Note that for SU(n) a possible maximum torus is a diagonal matrix with determinant 1.