Solution
= Solution
The map is a rotation about the $n$ axis exactly when
$$
a+b+c=1,\qquad \alpha^2+\gamma^2=1,
$$
or explicitly
$$
(2a-b-c)^2+3(b-c)^2=4.
$$
These conditions make the axis fixed and the perpendicular-plane action length preserving; they are also necessary.