Nonunital decompositions of the matrix algebra of order three

All decompositions of M3ℂ 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 M3ℂ into a direct vector-space sum of two subalgebras as well as description of Rota–Baxter operators of nonzero weight on M3ℂ is finished.