Eigenvalues of a $2 \times 2$ block matrix where every block is an identity matrix

Hint. Note that $A^2=2A$ and $\mbox{dim}(\mbox{Ker}(A))=n$.

Hint: Write it as $$( {\bf 1}_2 {{\bf 1}_2}^T )\otimes {\bf I}_n$$Then use the results how Kronecker products inherits their factor's eigenvalues.