  Suppose we wish to embed an (associative) $k$-algebra $A$ in a $k$-algebra
$R$ generated in some specified way; e.g., by two elements, or by copies of
given $k$-algebras $A_1,$ $A_2,$ $A_3.$ Several authors have obtained
sufficient conditions for such embeddings to exist. We prove here some further
results on this theme. In particular, we merge the ideas of existing
constructions based on two generating <i>elements</i>, and on three given
<i>subalgebra</i>, to get a construction using two given subalgebras.
  We pose some questions on how these results can be further strengthened.
