Answer by Glasby for Generalization of Schur's Lemma

Schur's lemma has a different generalization when the coefficient field $F$ is not algebraically closed. Then you get $M_{m_1}(D_1)\times\cdots\times M_{m_k}(D_k)$ where $D_i:={\rm Hom}_{FG}(\rho_i,\rho_i)$ is adivision algebra over $F$ by Schur's lemma. If $F$ is infinite, noncommutative $D_i$ can arise. For example, the quaternion group of order 8 has a 4-dimensional irreducible representation $\rho$ over the rationals, where ${\rm Hom}_{\mathbb{Q}G}(\rho,\rho)$ is the rational quaternions.