Let $G$ be a *simply connected* simple algebraic group over the field of complex numbers $\mathbb C$. Let $H$ be a *symmetric subgroup* of $G$. This means that there exists an automorphism of order 2 $\sigma\colon G\to G$ such that $H$ is the group of fixed points of $\sigma$ in $G$. It is known that $H$ is connected.

Let us assume that $H$ is *not semisimple*. It is known that then the center of $H$ is one-dimensional. Consider the derived subgroup $H^{\rm der}:=[H,H]$.

Question.Is it true that the semisimple group $H^{\rm der}$ is always simply connected?

It seems that this is true when $G$ is a classical group (say, when $G={\rm Sp}_{2n}$, $H=U_n$). What about $E_6$ and $E_7$? If the answer is always "yes", I would prefer a classification-free proof.