  A Majorana algebra is a commutative nonassociative real algebra generated by
a finite set of idempotents, called Majorana axes, that satisfy some of the
properties of the $2A$-axes of the Monster Griess algebra. The term was
introduced by A. A. Ivanov in 2009 inspired by the work of S. Sakuma and M.
Miyamoto. In the present paper, we revisit Mayer and Neutsch's theorem on
associative subalgebras of the Griess algebra in the context of Majorana
theory. We apply this result to determine all the maximal associative
subalgebras of some low-dimensional Majorana algebras; namely, the Majorana
algebras generated by two Majorana axes and the Majorana representations of the
symmetric group of degree $4$ involving $3C$-algebras.
