The exponential mapping takes the Lie algebra of the Lorentz group into the Lorentz group. Each element of the group is defined as a formal power series, while the product of two exponential elements usually involves an infinite sum of commutator terms, such as in the Baker–Campbell–Hausdorff formula. Because of the special arithmetic in Clifford algebras, many Baker–Campbell–Hausdorff‐like formulas and identities can be calculated or summed exactly when they involve only elements from the algebra. We calculate exact identities for the Baker–Campbell–Hausdorff formula and related formulas in the quaternion and dihedral algebras. These are useful in treatments of the Lorentz group, and make possible a truly finite (as opposed to infinitesimal) description of a transformation group in physics.