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This paper focuses on Euler angles and on the decomposition of rotations. We consider arbitrary rotation axes that are not necessarily mutually orthogonal; we characterize the set of rotation matrices that admit Euler angles about arbitrary rotation axes; and we provide a single set of Euler angle formulas that applies to any selection of rotation axes. The results are presented and derived in a coordinate-free setting, where no reference frames are required, and no components of any array or matrix are manipulated.