$a$ sum of $2$ squares and $b$ sum of $3$ squares imply $a^2b$ is sum of $3$ squares.

You do not need any property for $a$, since its square is a square.

If $b=v^2+w^2+z^2$ (copying Simon Fox's description) then

$$a^2b =a^2v^2+a^2w^2+a^2z^2 = (av)^2+(aw)^2+(az)^2$$