I am sorry if this question is too elementary to be posted here, but no experts answer this question when I post it on Math Stackexchange.

Let $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ be a Cartan decomposition for a noncompact real simple Lie algebra $\mathfrak{g}$ corresponding to a Cartan involution $\theta$, where $\mathfrak{k}$ is the maximal compact subalgebra of $\mathfrak{g}$. Suppose that $\sigma$ is another involutive automorphism of $\mathfrak{g}$ such that $\sigma\theta=\theta\sigma$. Then $\sigma$ preserves the Cartan decomposition, and $\sigma|_\mathfrak{k}:\mathfrak{k}\rightarrow\mathfrak{k}$. By the classification for symmetric pairs, it seems true that $\sigma|_\mathfrak{k}$ is never the identity map, but how to prove this fact theoretically (instead of case by case)?

In other words, there is no (non-Cartan) involutive automorphism $\sigma$ of a noncompact real simple Lie algebra $\mathfrak{g}$ such that the subalgebra $\mathfrak{g}^\sigma$ of the fixed points under the action of $\sigma$ on $\mathfrak{g}$ contains a maximal compact subalgebra. How to prove it? I shall be grateful if experts here may offer any hint.