Hausdorff Paradox/Lemma 1/Proof Outline

Proof Outline
We determine the angle $\theta$ between $a_\phi$ and $a_\psi$ such that no element of $G$ other than its identity $\mathbf I_3$ represents the identity rotation.

Let us consider a typical element $\alpha$ of $G$:
 * $\alpha = \phi \circ \psi^{\pm 1} \circ \cdots \circ \phi \circ \psi^{\pm 1}$

Using the properties of orthogonal transformations and elementary trigonometry, we seek to prove that the equation:
 * $\alpha = \mathbf I_3$

has only finitely many solutions.

Consequently, except for a countable set of values, we may select any angle $\theta$ that satisfies the requirements.