Let $T\colon\mathbb{R}^3\to\mathbb{R}^3$ be an orthogonal linear transformation so that $\det T = 1$ and $T$ is not the identity transformation. Consider $S = \{ ( x , y , z ) : x^2 + y^2 + z^2 = 1 \}$. Show that $T$ fixes exactly $2$ points on $S$.
I have proceeded like this. $T\colon \mathbb{R}^3\to\mathbb{R}^3$ implies $T$ has at least one real eigenvalue (as the characteristic polynomial is of degree $3$). We know this real eigenvalue must be $1$ or $-1$. Then I am trying to show it has the eigenvalue $1$ of multiplicity $1$. Thanks for any help.