## (a^2 + b^2 + c^2 + d^2)(p^2 + q^2 + r^2 + s^2) — Part 3

$(a^2 + b^2 + c^2 + d^2) \,(p^2 + q^2 + r^2 + s^2)$

$= \; a^2 \, p^2+a^2 \, q^2+a^2 \, r^2+a^2 \, s^2+b^2 \, p^2+b^2 \, q^2+b^2 \, r^2+b^2 \, s^2+c^2 \, p^2$
$+ \; c^2 \, q^2+c^2 \, r^2+c^2 \, s^2+d^2 \, p^2+d^2 \, q^2+d^2 \, r^2+d^2 \, s^2$

$= \; (ap - bq - cr - ds)^2 + (aq + bp + cs - dr)^2 + (ar + cp + dq - bs)^2 + (as + dp + br - cq)^2$

[ a sum of four squares]

Or

$(a \,p+b \,q+c \,r+d \,s)^2+(a \,q-b \,p)^2+(a \,r-c \,p)^2+(a \,s-d \,p)^2+(b \,r-c \,q)^2$
$+ \; (b \,s-d \,q)^2+(c \,s-d \,r)^2$

[ as a sum of seven squares ]

Can you find positive integers   $a, b, c, d, p, q, r, s$   so that the expressions

$a \, p \; - \; b \, q \; - \; c \, r \; - \; d \, s$
$a \, q \; + \; b \, p \; + \; c \, s \; - \; d \, r$
$a \, r \; + \; c \, p \; + \; d \, q \; - \; b \, s$
$a \, s \; + \; d \, p \; + \; b \, r \; - \; c \, q$