  In a previous article, [arXiv:1501.02506, JPhysA {\bf48} (2015) 225207], we
demonstrated that whenever $[X,Y] = u X + vY + cI$ 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 \[
f(u,v) = {(u-v)e^{u+v}-(ue^u-ve^v)\over u v (e^u - e^v)} =
{(u-v)-(ue^{-v}-ve^{-u})\over u v (e^{-v} - e^{-u})}. \] This is much more
general than the results usually presented for either the Heisenberg commutator
$[P,Q]=-i\hbar I$, or the creation-destruction commutator $[a,a^\dagger]=I$. 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 \[ \ln( e^X e^Y ) = X + Y + {I \over e^{-L_X} - e^{+L_Y} } \left(
{I-e^{-L_X}\over L_X} + {I-e^{+L_Y}\over L_Y} \right) [X,Y]. \]
