The authors extend some results of L. Makar-Limanov and P. Malcolmson [Proc. Am. Math. Soc. 111, No. 2, 315–322 (1991; Zbl 0719.16008)] and Z. Reichstein [Proc. Am. Math. Soc. 124, No. 1, 17–19 (1996; 0847.16015)]. Let \(K\) be a field and \(\mathfrak{M}\) a homogeneous variety of (not necessarily associative) \(K\)-algebras. The authors say that \(\mathfrak{M}\) is an MLM variety if for any field extension \(F\) of \(K\), any algebra \(A\in\mathfrak{M}_F\), and any subset of at least two elements \(Y\) in \(A\) such that the \(K\)-subalgebra of \(A\) generated by \(Y\) is the free \(K\)-algebra in the variety \(\mathfrak{M}\) freely generated by \(Y\), the \(F\)-subalgebra of \(A\) generated by \(Y\) is the free \(F\)-algebra in the variety \(\mathfrak{M}_F\) freely generated by \(Y\). Additionally assuming that the field \(K\) is uncountable, the authors say that \(\mathfrak{M}\) is Reichstein variety if for any algebra \(A\in\mathfrak{M}_F\) and any field extension \(F\) of \(K\) such that \(F\otimes_KA\) contains a free \(K\)-algebra in the variety \(\mathfrak{M}\)on at least two free generators, \(A\) contains a free \(K\)-algebra in the variety \(\mathfrak{M}\) on the same number of free generators. In this terminology, the results of the two above-cited papers say that the variety of all associative \(K\)-algebras is both MLM and Reichstein. The authors prove that if \(K\) is an uncountable field, then every MLM variety of \(K\)-algebras is Reichstein. Then they show that the following varieties are MLM: the variety of all \(K\)-algebras, the variety of all commutative \(K\)-algebras, the variety of all Lie \(K\)-algebras, the variety of all anticommutative \(K\)-algebras, and, provided that \(\mathrm{char }K\ne 2\), the variety generated by all special Jordan \(K\)-algebras. Similar results hold for the variety of all non-commutative Poisson \(K\)-algebras.