Characterises elements of $\mathrm{SO}_3(\mathbb{R})$ geometrically: every rotation in three dimensions fixes an axis and rotates about it. The proof uses the fact that $\det(A - I) = 0$ for any $A \in \mathrm{SO}_3(\mathbb{R})$ (an eigenvalue-$1$ argument exploiting the odd dimension), then constructs an orthonormal basis in which the matrix takes the standard rotation form.