Prove Sum of Four Squares Identity (8.10).
The Sum of Four Squares Identity (8.10) is
$$ \begin{align} (a^2 + b^2 + c^2 + d^2) \times (x^2 + y^2 + z^2 + w^2) & = (ax + by + cz + dw)^2 + (az− bw− cx + dy)^2 \\ \\ & + (ay− bx + cw− dz)^2 + (aw + bz− cy− dx)^2 \end{align} $$
We'll expand the LHS first:
$$\begin{align} (a^2 + b^2 + c^2 + d^2) \times (x^2 + y^2 + z^2 + w^2) & = a^2 w^2 + a^2 x^2 + a^2 y^2 + a^2 z^2 \\ & + b^2 w^2 + b^2 x^2 + b^2 y^2 + b^2 z^2 \\ & + c^2 w^2 + c^2 x^2 + c^2 y^2 + c^2 z^2 \\ & + d^2 w^2 + d^2 x^2 + d^2 y^2 + d^2 z^2 \end{align}$$
We'll expand the RHS first:
$$\begin{align} (ax + by + cz + dw)^2 + (az− bw− cx + dy)^2 + \\ (ay− bx + cw− dz)^2 + (aw + bz− cy− dx)^2 & = a^2 x^2 + 2 a b x y + 2 a c x z + 2 a d w x \\ & + b^2 y^2 + 2 b c y z + 2 b d w y \\ &+ c^2 z^2 + 2 c d w z + d^2 w^2 \\ & + a^2 z^2 - 2 a b w z - 2 a c x z + 2 a d y z \\ & + b^2 w^2 + 2 b c w x - 2 b d w y \\ & + c^2 x^2 - 2 c d x y + d^2 y^2 \\ & + a^2 y^2 - 2 a b x y + 2 a c w y - 2 a d y z \\ &+ b^2 x^2 - 2 b c w x + 2 b d x z \\ & + c^2 w^2 - 2 c d w z + d^2 z^2 \\ & + a^2 w^2 + 2 a b w z - 2 a c w y - 2 a d w x \\ & + b^2 z^2 - 2 b c y z - 2 b d x z \\ & + c^2 y^2 + 2 c d x y + d^2 x^2 \\ \\ & = a^2 w^2 + a^2 x^2 + a^2 y^2 + a^2 z^2 \\ & + b^2 w^2 + b^2 x^2 + b^2 y^2 + b^2 z^2 \\ & + c^2 w^2 + c^2 x^2 + c^2 y^2 + c^2 z^2 \\ & + d^2 w^2 + d^2 x^2 + d^2 y^2 + d^2 z^2 \end{align}$$