= Solution
The <orthogonal group> is
$$
O(n)=\{Q\in M_n(\mathbb R):Q^TQ=I\}.
$$
The <special orthogonal group> is its determinant-one subgroup
$$
SO(n)=\{Q\in O(n):\det Q=1\}.
$$
Every <eigenvalue> of an <orthogonal matrix> has <modulus> one. A real three-by-three matrix has at least one real eigenvalue, and nonreal eigenvalues occur in a <complex conjugate> pair. If $Q\in SO(3)$ has such a pair $e^{\pm i\theta}$, their product is one, so the remaining eigenvalue is $\det Q=1$. If all three eigenvalues are real, each is $\pm1$ and their product is one; an odd number of three signs with product one must include $+1$. Hence every element of $SO(3)$ has an eigenvector of eigenvalue one and represents a <rotation in three dimensions> about its span.
\b[It is false] that every element of $O(3)$ is either a rotation or a plane reflection. For example,
$$
Q=\begin{pmatrix}
\cos\theta&-\sin\theta&0\\
\sin\theta&\cos\theta&0\\
0&0&-1
\end{pmatrix},
\qquad 0<\theta<\pi,
$$
is an <improper orthogonal transformation> combining a rotation with a reflection. Its eigenvalues are $e^{\pm i\theta}$ and $-1$, whereas a plane reflection has eigenvalues $1,1,-1$ and a rotation has determinant one.
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