Product of Sums of Four Squares/Proof 2

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Theorem

Let $a, b, c, d, w, x, y, z$ be numbers.

Then:

\(\ds \) \(\) \(\ds \left({a^2 + b^2 + c^2 + d^2}\right) \left({w^2 + x^2 + y^2 + z^2}\right)\)
\(\ds =\) \(\) \(\ds \left({a w + b x + c y + d z}\right)^2\)
\(\ds \) \(+\) \(\ds \left({a x - b w + c z - d y}\right)^2\)
\(\ds \) \(+\) \(\ds \left({a y - b z - c w + d x}\right)^2\)
\(\ds \) \(+\) \(\ds \left({a z + b y - c x - d w}\right)^2\)


Proof

Let:

$\mathbf m = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$
$\mathbf n = -w \mathbf 1 + x \mathbf i + y \mathbf j + z \mathbf k$

be two quaternions.

Then:

\(\ds \size {\mathbf m} \size {\mathbf n}\) \(=\) \(\ds \size {\mathbf m \mathbf n}\) Quaternion Modulus of Product of Quaternions
\(\ds \leadsto \ \ \) \(\ds \paren {a^2 + b^2 + c^2 + d^2} \paren {\paren {-w}^2 + x^2 + y^2 + z^2}\) \(=\) \(\ds \size {\mathbf m \mathbf n}\) Definition of Quaternion Modulus
\(\ds \) \(=\) \(\ds \paren {-a w - b x - c y - d z}^2\) Definition of Quaternion Multiplication
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {a x - b w + c z - d y}^2\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {a y - b z - c w + d x}^2\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {a z + b y - c x - d w}^2\)
\(\ds \leadsto \ \ \) \(\ds \paren {a^2 + b^2 + c^2 + d^2} \paren {w^2 + x^2 + y^2 + z^2}\) \(=\) \(\ds \paren {a w + b x + c y + d z}^2\) simplifying
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {a x - b w + c z - d y}^2\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {a y - b z - c w + d x}^2\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {a z + b y - c x - d w}^2\)

$\blacksquare$


Sources