Prove that isometry in $\mathbb R^3$ can be written as a composition of at most three reflections across some planes

Question

Let $t$ be an isometry written as $t(x) = Qx$ where $Q$ is orthogonal $3x3$ matrix and $x$ some 3-dimensional vector. Prove that $t$ can be written as a composition of at most three reflections across some planes.

I guess its a simple proof but I have no idea how to start. Can you help?

Answer 1

Start with the reflection that, maps $e_1$ to $t(e_1)$. Then take the reflection that maps the reflection of $e_2$ to $t(e_2)$ and fixes(!) $t(e_1)$. The either $e_3$ is already mapped to $t(e_3)$, or you need to add a reflection that fixes $t(e_1)$ and $t(e_2)$ and necessarily maps $e_3$ as desired.