Variables (n : nat) (rG : mx_representation G n).
Definition gring_mx := vec_mx \o mulmxr (enveloping_algebra_mx rG).
Canonical gring_mx_linear := [linear of gring_mx].

(* Specifications: *)
Lemma gring_mxJ a x :
x \in G -> gring_mx (a *m aG x) = gring_mx a *m rG x.