We classify algebraic curvature tensors such that the Ricci operator $\rho$ is simple (i.e. $\rho$ is complex diagonalizable and $Spec\left\{\rho\right\=\left\{a\right\}$ or $Spec\left\{\rho\right\}=\left\{a_{1}\pm a_{2}\sqrt{-1}\right\}$) and which are Jacobi–Ricci commuting (i.e. $\rhoJ(v)=J(v)\rho$ for all $v$).