The Minimum Number of Rotations About Two Axes for Constructing an Arbitrary Fixed Rotation

For any pair of three-dimensional real unit vectors $\hat{m}$ and $\hat{n}$ with $|\hat{m}^{\rm T} \hat{n}| < 1$ and any rotation $U$, let $N_{\hat{m},\hat{n}}(U)$ denote the least value of a positive integer $k$ such that $U$ can be decomposed into a product of $k$ rotations about either $\hat{m}$ or $\hat{n}$. This work gives the number $N_{\hat{m},\hat{n}}(U)$ as a function of $U$. Here a rotation means an element $D$ of the special orthogonal group ${\rm SO}(3)$ or an element of the special unitary group ${\rm SU}(2)$ that corresponds to $D$. Decompositions of $U$ attaining the minimum number $N_{\hat{m},\hat{n}}(U)$ are also given explicitly.