Euclidean vector space/R^3/Proper isometry/Representation/Fact

Let

$\varphi \colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{3}$

be a proper isometry.

Then ${}\varphi$ is a rotation around a fixed axis.

This means that ${}\varphi$ is described, with respect to a suitable orthonormal basis, by a matrix of the form

${\begin{pmatrix}1&0&0\\0&\operatorname {cos} \,\alpha &-\operatorname {sin} \,\alpha \\0&\operatorname {sin} \,\alpha &\operatorname {cos} \,\alpha \end{pmatrix}}.$

Proof, Write another proof