  All decompositions of $M_3(\mathbb{C})$ into a direct vector-space sum of two
subalgebras such that none of the subalgebras contains the identity matrix are
classified. Thus, the classification of all decompositions of $M_3(\mathbb{C})$
into a direct vector-space sum of two subalgebras as well as description of
Rota-Baxter operators of nonzero weight on $M_3(\mathbb{C})$ is finished.
