The minimum number of rotations about two axes for constructing an arbitrarily fixed rotation

For any pair of three-dimensional real unit vectors [Formula: see text] and [Formula: see text] with [Formula: see text] and any rotation U, let [Formula: see text] denote the least value of a positive integer k such that U can be decomposed into a product of k rotations about either [Formula: see text] or [Formula: see text]. This work gives the number [Formula: see text] as a function of U. Here, a rotation means an element D of the special orthogonal group SO(3) or an element of the special unitary group SU(2) that corresponds to D. Decompositions of U attaining the minimum number [Formula: see text] are also given explicitly.