Johnson describes a method for finding $3\times3\times23$ matrix multiplication algorithms. One of the tools he uses is DeGroote's sandwiching equations, although the exact steps are not detailed. Here's the problem.
We are given three known $3\times3$ real valued matrices, $A$, $B$ and $C$. In DeGroote's context these matrices are equivalent to
$\overline{A}=PAQ^-1$,
$\overline{B}=QBR^-1$ and
$\overline{C}=RCP^-1$, respectively,
where $P$, $Q$ and $R$ are invertible $3\times3$ matrices. Johnson used this 'sandwiching' transformation to obtain suitable (i.e. nicer) $\overline{A}, \overline{B}$ and $\overline{C}$ matrices.
How did Johnson do this?
If $A, \overline{A}, B, \overline{B},C$ and $\overline{C}$ are given, how one finds $P, Q$ and $R$?