Suppose I have a $4 \times $ matrix $M$ which I will write in terms of blocks as follows: $$M = \begin{pmatrix} A & B \\ C & D \end{pmatrix} $$ I know that $M$ has an eigenvalue $\lambda = 0$ with two associate eigenvectors, say $v_{1}$ and $v_{2}$. In addition, $M$ has two other eigenvectors $v_{3}$ and $v_{4}$ associated to two different eigenvalues. Is it possible to write $M$ as a direct sum of matrices: $$M = M_{1}\oplus M_{2}$$ where $M_{1}$ is a linear map on the subspace generated by $v_{1}$ and $v_{2}$ and $M_{2}$ is a linear map on the subspace generated by $v_{3}$ and $v_{4}$? If so, is there a recipe to construct such direct sum explicitly?
Note: The matrix $M$ is assumed as a linear map on $\mathbb{C}^{4}$.