Es ist

${\displaystyle {}{\left(a+b{\sqrt {p}}+c{\sqrt {q}}+d{\sqrt {pq}}\right)}\cdot {\sqrt {p}}=a{\sqrt {p}}+pb+c{\sqrt {pq}}+pd{\sqrt {q}}\,,}$

${\displaystyle {}{\left(a+b{\sqrt {p}}+c{\sqrt {q}}+d{\sqrt {pq}}\right)}\cdot {\sqrt {q}}=a{\sqrt {q}}+b{\sqrt {pq}}+qc+qd{\sqrt {p}}\,}$

und

${\displaystyle {}{\left(a+b{\sqrt {p}}+c{\sqrt {q}}+d{\sqrt {pq}}\right)}\cdot {\sqrt {pq}}=a{\sqrt {pq}}+pb{\sqrt {q}}+qc{\sqrt {p}}+pqd\,.}$

Somit ist die Multiplikatiosmatrix gleich

${\displaystyle {\begin{pmatrix}a&pb&qc&pqd\\b&a&qd&qc\\c&pd&a&pb\\d&c&b&a\end{pmatrix}}.}$