Non-semisimple symmetric subgroups of simply connected simple algebraic groups 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.
 A: In your case, $H$ is a Levi subgroup and the derived subgroup of a Levi subgroup in a simply connected group is always simply connected (since all fundamental weights are characters).
A: The answer is YES. Such a subgroup $H$ is called a Hermitian symmetric subgroup (because then for suitable real forms of $G$ and $H$, the quotient $G/H$ is a Hermitian symmetric space). It follows from the theory of Victor Kac that any Hermitian symmetric subgroup of a simple group is a "diagrammatic" subgroup (Levi subgroup) corresponding to the Dynkin subdiagram obtained by removing one vertex corresponding to a simple root appearing with coefficient one in the expression of the highest root as a linear combination of simple roots. See e.g. Table 7, type II in the book by Onishchik and Vinberg, or  case (B) in the text after Lemma 5.17 on page 513 in Ch. X of the book by Helgason "Differential Geometry, Lie Groups, and Symmetric Spaces".  It is well known that the derived group of any Levi subgroup of a simply connected semisimple group is simply connected, see e.g. Proposition 12.14 in the book by G. Malle and D. Testerman "Linear Algebraic Groups and Finite Groups of Lie Type".
