= Solution
A real three-by-three <matrix> represents a proper <rotation> in the standard orthonormal basis precisely when $A^TA=I$ and $\det A=1$, that is, when $A\in\mathrm{SO}(3)$. For the supplied integer numerator $B=3A$, direct multiplication gives $B^TB=9I$ and $\det B=27$, verifying both conditions.
The rotation axis is the <eigenspace> for <eigenvalue> one. Solving $(A-I)n=0$ gives $n_3=0$ and $n_2=2n_1$, so
$$
\boxed{n=\frac1{\sqrt5}(1,2,0)^T}
$$
is a unit axis vector; its negative describes the same axis. On its perpendicular plane the <rotation matrix> has eigenvalues $e^{i\theta}$ and $e^{-i\theta}$. The <rotation trace formula> therefore gives $\operatorname{tr}A=1+2\cos\theta$. Since the trace is $-1/3$,
$$
\boxed{\cos\theta=-\frac23.}
$$
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