= Solution
A three-dimensional <rotation matrix> is an <orthogonal matrix> with <determinant> $+1$. Here $\det A=b(a^2-d^2)$. For the branch $d=0$, the determinant condition gives $b=1$; for the branch $a=0$, it gives $b=-1$. Hence
$$
\boxed{(a,b,c)^{\mathsf T}\in\{(1,1,1)^{\mathsf T},(-1,1,1)^{\mathsf T},(0,-1,0)^{\mathsf T},(0,-1,2)^{\mathsf T}\}.}
$$
The first is the identity <rotation>; the remaining three are half-turns. Their fixed axes are respectively the $y$ axis, the direction $(1,0,-1)$ and the direction $(1,0,1)$. The identity is included under the usual <rotation matrix> convention.
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