Nonmatrix varieties of nonassociative algebras

A variety of associative algebras is nonmatrix if it does not contain the algebra of $2\times 2$ matrices over a given field. Nonmatrix varieties were introduced and studied by V. N. Latyshev in [Algebra and Logic, 16, No. 2, 98—122 (1977); Algebra and Logic, 16, No. 2, 122—133 (1977); Mat. Zam., 27, No. 1, 147—156 (1980)] in connection with the Specht problem. A series of equivalent characterizations of nonmatrix varieties was obtained in [Isr. J. Math., 181, No. 1, 337—348 (2011)]. In the present paper, the notion of nonmatrix variety is extended to nonassociative algebras, and their characterization from the last-mentioned paper is generalized to alternative, Jordan, and some other varieties of algebras.