  The quasi-Clifford algebras, and their Wedderburn structure and
representation theory, as described by Gastineau-Hills in 1980 and 1982, should
be better known, and have only recently been rediscovered. These algebras and
their representation theory provide effective tools to address certain
questions relating to plug-in constructions for Hadamard matrices. The key
question addressed is: Given $\lambda$, a pattern of amicability /
anti-amicability, with $\lambda_{j,k}=\lambda_{k,j}=\pm 1$, find a set of $n$
monomial $\{-1,0,1\}$ matrices $\{D_j\}$ of minimal order such that $$ D_j
D_k^T - \lambda_{j,k} D_k D_j^T = 0 \quad (j \neq k). $$
