Аннотация:We consider the bilinear complexity of multiplication in local and semisimple algebras over an infinite field of characteristic differing from 2. We obtain a criterion for the rank of a local algebra to be almost minimal. We evaluate the bilinear complexity of the algebras of generalised quaternions over such a field; we prove that any simple algebra of almost minimal rank is an algebra of generalised quaternions. This result is used for the classification of semisimple algebras of almost minimal rank.