  For a cocycle of invertible real $n$-by-$n$ matrices, the Multiplicative
Ergodic Theorem gives an Oseledets subspace decomposition of $\mathbb{R}^n$;
that is, above each point in the base space, $\mathbb{R}^n$ is written as a
direct sum of equivariant subspaces, one for each Lyapunov exponent of the
cocycle. It is natural to ask if these summands may be further decomposed into
equivariant subspaces; that is, if the Oseledets subspaces are reducible. We
prove a theorem yielding sufficient conditions for irreducibility of the
trivial equivariant subspaces $\mathbb{R}^2$ and $\mathbb{C}^2$ for
$O_2(\mathbb{R})$-valued cocycles and give explicit examples where the
conditions are satisfied.
