An advance in the arithmetic of the Lie groups as an alternative to the forms of the Campbell-Baker-Hausdorff-Dynkin theorem

The exponential of an operator or matrix is widely used in quantum theory, but it sometimes can be a challenge to evaluate. For non-commutative operators $\mathbf{X}$ and $\mathbf{Y}$, according to the Campbell-Baker-Hausdorff-Dynkin theorem, ${\mathrm{e}}^{\mathbf{X}+\mathbf{Y}}$ is not equivalent to ${\mathrm{e}}^{\mathbf{X}}{\mathrm{e}}^{\mathbf{Y}}$, but is instead given by the well-known infinite series formula. For a Lie algebra of a basis of three operators $\{\mathbf{X},\mathbf{Y},\mathbf{Z}\}$, such that $[\mathbf{X},\mathbf{Y}]=\kappa \mathbf{Z}$ for scalar $\kappa$ and cyclic permutations, here it is proven that ${\mathrm{e}}^{a\mathbf{X}+b\mathbf{Y}}$ is equivalent to ${\mathrm{e}}^{p\mathbf{Z}}{\mathrm{e}}^{q\mathbf{X}}{\mathrm{e}}^{-p\mathbf{Z}}$ for scalar $p$ and $q$. Extensions for ${\mathrm{e}}^{a\mathbf{X}+b\mathbf{Y}+c\mathbf{Z}}$ are also provided. This method is useful for the dynamics of atomic and molecular nuclear and electronic spins in constant and oscillatory transverse magnetic and electric fields.