Space isometry/Matrix/From 4 numbers/Explicit/Exercise

Show that for ${}a,b,c,d\in \mathbb {R}$ satisfying ${}a^{2}+b^{2}+c^{2}+d^{2}=1$, the matrix

${\begin{pmatrix}a^{2}+b^{2}-c^{2}-d^{2}&2(-ad+bc)&2(ac+bd)\\2(ad+bc)&a^{2}-b^{2}+c^{2}-d^{2}&2(-ab+cd)\\2(-ac+bd)&2(ab+cd)&a^{2}-b^{2}-c^{2}+d^{2}\end{pmatrix}}$

defines an isometry on ${}\mathbb {R} ^{3}$.

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