Does the Kronecker product preserve common irreducibility A set of $d\times d$ real or complex matrices is called “commonly irreducible” if those matrices do not jointly preserve a linear subspace with dimension strictly between $0$ and $d$.
I wanted to know whether the Kronecker product preserves this property. In other words, given two sets of commonly irreducible matrices $S$ and $Q$, is the set
$ 
S\otimes Q = \{P_1 \otimes P_2 \mid P_1 \in S, P_2\in Q\} 
$
commonly irreducible as well?
 A: The question can be rephrased as follows:

Let $$ be a field.
Let $M$ and $N$ be finite-dimensional simple modules over $$-algebras $A$ and $B$ respectively.
Is $M ⊗_ N$ simple as an $A ⊗_ B$ module?

This question is answered in Tensor products of simple modules over algebras.
The assertion is true if $$ is algebraically closed (a consequence of Burnside’s theorem on matrix algebras), but false if $$ is not algebraically closed.
So yes, over $ℂ$ the set $S ⊗ Q$ will be commonly irreducible again.
Over $ℝ$ we can use the linked question to find an explicit counterexample:
the rotation matrix
$$
  J ≔
  \begin{pmatrix}
    0 &           -1 \\ 
    1 & \phantom{-}0
  \end{pmatrix}
$$
preserves no one-dimensional linear subspace of $ℝ^2$, but the matrix
$$
  J ⊗ J
  =
  \begin{pmatrix}
    0 &           -J \\ 
    J & \phantom{-}0
  \end{pmatrix}
  =
  \begin{pmatrix}
    0 & \phantom{-}0 & \phantom{-}0 & 1 \\
    0 & \phantom{-}0 &           -1 & 0 \\
    0 &           -1 & \phantom{-}0 & 0 \\
    1 & \phantom{-}0 & \phantom{-}0 & 0
  \end{pmatrix}
$$
will have to preserve some proper, non-zero linear subspaces of $ℝ^4$.
We can actually read off a decomposition of $ℝ^4$ into one-dimensional linear subspaces that are preserved by $J ⊗ J$, namely
$$
  ℝ^4
  =
  \langle e_1 + e_4 \rangle
  ⊕ \langle e_1 - e_4 \rangle
  ⊕ \langle e_2 + e_3 \rangle
  ⊕ \langle e_2 - e_3 \rangle \,.
$$
