= Solution
Directly multiplying the columns' <scalar products> gives $C^TC=I$, and direct evaluation gives $\det C=1$. Thus \b[the operation is a <proper rotation>]. Its <trace> is zero. A three-dimensional <proper rotation> has <eigenvalues> $1,e^{i\varphi},e^{-i\varphi}$, hence $\operatorname{tr}C=1+2\cos\varphi$. Therefore
$$
\boxed{\varphi=\frac{2\pi}{3}=120^\circ.}
$$
This determines the usual unsigned angle in $[0,\pi]$ without changing <coordinates>. The axis can be checked, if desired, from $C(0,1,-1)^T=(0,1,-1)^T$. The <determinant> test must precede the <trace> formula: an <improper rotation> instead has <trace> $-1+2\cos\varphi$.
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