Explicit Baker-Campbell-Hausdorff formulae for some specific Lie algebras

In a previous article, [arXiv:1501.02506, JPhysA {\bf48} (2015) 225207], we demonstrated that whenever $[X,Y]=uX+vY+cI$ the Baker-Campbell-Hausdorff formula reduces to the tractable closed-form expression $$Z(X,Y)=\ln (e^X{e}^Y)=X+Y+f(u,v)[X,Y],$$ where $f(u,v)=f(v,u)$ is explicitly given by $$f(u,v)=\frac{(u-v)e^{u+v}-(u{e}^u-v{e}^v)}{uv(e^u-e^v)}=\frac{(u-v)-(u{e}^{-v}-v{e}^{-u})}{uv(e^{-v}-e^{-u})}\mathrm{.}$$ This is much more general than the results usually presented for either the Heisenberg commutator $[P,Q]=-i\mathrm{\hslash }I$, or the creation-destruction commutator $[a,a^{†}]=I$. In the current article we shall further generalize and extend this result, primarily by relaxing the input assumptions. We shall work with the structure constants $f_{ab}{}^c$ of the Lie algebra, (defined by $[T_a,T_b] = f_{ab}{}^c ; T_c$), and identify suitable constraints one can place on the structure constants to make the Baker–Campbell–Hausdorff formula tractable. We shall also develop related results using the commutator sub-algebra $[\mathfrak{g},\mathfrak{g}]$ of the relevant Lie algebra $\mathfrak{g}$. Under suitable conditions, and taking $L_{A}B=[A,B]$ as usual, we shall demonstrate that $$\ln (e^X{e}^Y)=X+Y+\frac{I}{e^{-L_X}-e^{+L_Y}}(\frac{I-e^{-L_X}}{L_X}+\frac{I-e^{+L_Y}}{L_Y})[X,Y]\mathrm{.}$$