Let $P\subset\Bbb R^d$ be a *vertex-transitive polytope* aka. an *orbit polytope*.
Can there be a matrix $T\in\mathrm{SO}(\Bbb R^d)$ that commutes with all symmetries in $\mathrm{Aut}(P)\subset\mathrm O(\Bbb R^d)$?

Probably one approach to the question is as follows: can there be a vertex-transitive polytope $P\subset\Bbb R^d$ for which $\mathrm{Aut}(P)$ is (real) irreducible, but **not** absolutely irreducible (that is, not irreducible over $\Bbb C$).


Vertex-transitivity is necessary for all these questions. For example, there is a polytope (not vertex-transitive) whose symmetry group is a finite subgroup of $\mathrm{SO}(\Bbb R^2)$, which is real irreducible, but reducible over $\Bbb C$.
Since $\mathrm{SO}(\Bbb R^2)$ is commutative, every element of that group would then commute with $\mathrm{Aut}(P)$.
It is known that most commutative groups cannot be symmetry groups of vertex-transitive polytopes (only exceptions are elementary 2-abelian groups).