The centralizer of a $3$-dimensional simple subalgebra in the universal enveloping algebra of a $7$-dimensional simple Malcev algebra

Under study are the centralizers of $3$-dimensional simple Lie subalgebras in the universal enveloping algebra of a $7$-dimensional simple Malcev algebra. We find some sets of generators for these centralizers in characteristic not 2 nor 3 and for the subalgebra generated by the centralizer in the central closure of the universal enveloping algebra in characteristic 3. As a corollary of the main theorem we obtain the available description of the center of universal enveloping algebra of a $7$-dimensional simple Malcev algebra.