Rather than overload the question with side remarks, I'll add some background here in community-wiki format to indicate what can be gotten from Springer's relatively elementary treatment of regular elements here. He deals in great generality with finite complex (=unitary) reflection groups, including all finite Coxeter groups (= real reflection groups), the Weyl groups being the most important of these. Besides the structure and classification results of Coxeter and Shephard-Todd, he relies mainly on Chevalley's treatment of invariants in the standard matrix realization.

Folding of Coxeter diagrams comes up just in the case of the Weyl groups $W$ which I listed (though less frequently for $A_n$ when $n$ is even). Lie theory and the crystallographic root systems aren't really needed for my question, except perhaps for type $D_n$ with $n$ even. Consider the very special case of regular elements $w$ of order $d=2$ in Springer's paper, which exist and are all conjugate. Write the list of degrees of fundamental invariants as $d_1, \dots, d_n$. Of these the even ones (those divisible by $d$) are the degrees of the centralizer of $w$ in $W$, itself a finite (real or complex) reflection group.

Leaving aside the case of $D_n$ for even $n$, my list of types matches those $W$ not containing $-1$ as longest element $w_\circ$. On the other hand, it's easy to see that $w_\circ$ is always regular (For any finite Coxeter group). Since $-1 \in W$ iff all $d_i$ are even, the centralizer of our regular $w_\circ$ is a proper subgroup having the degrees of the Coxeter group obtained by folding and is in fact that subgroup (easy to check how the eigenvalues $\pm 1$ behave).

As in my question, the isolated case $E_6 \rightsquigarrow F_4$ gives a nice example of the resulting subgroup embedding. Have these embeddings been written down explicitly?

Rather than overload the question with side remarks, I'll add some background here in community-wiki format to indicate what can be gotten from Springer's relatively elementary treatment of regular elements here. ("Regular elements of finite reflection groups", Invent. Math. 25 (1974), 159-198.)

He deals in great generality with finite complex (= unitary) reflection groups, including all finite Coxeter groups (= real reflection groups), the Weyl groups being the most important of these. Besides the structure and classification results of Coxeter and Shephard-Todd, he relies mainly on Chevalley's treatment of invariants in the standard matrix realization.

Folding of Coxeter diagrams comes up just in the case of the Weyl groups $W$ which I listed (though less frequently for $A_n$ when $n$ is even). Lie theory and the crystallographic root systems aren't really needed for my question, except perhaps for type $D_n$ with $n$ even. Consider the very special case of regular elements $w$ of order $d=2$ in Springer's paper, which exist and are all conjugate. Write the list of degrees of fundamental invariants as $d_1, \dots, d_n$. Of these the even ones (those divisible by $d$) are the degrees of the centralizer of $w$ in $W$, itself a finite (real or complex) reflection group.

Leaving aside the case of $D_n$ for even $n$, my list of types matches those $W$ not containing $-1$ as longest element $w_\circ$. On the other hand, it's easy to see that $w_\circ$ is always regular (for any finite Coxeter group). Since $-1 \in W$ iff all $d_i$ are even, the centralizer of our regular $w_\circ$ is a proper subgroup having the degrees of the Coxeter group obtained by folding and is in fact that subgroup (easy to check how the eigenvalues $\pm 1$ behave).

As in my question, the isolated case $E_6 \rightsquigarrow F_4$ gives a nice example of the resulting subgroup embedding. Have these embeddings been written down explicitly?