  We show that there are {\it 13 types} of commutator algebras leading to the
new closed forms of the Baker-Campbell-Hausdorff (BCH) formula
$$\exp(X)\exp(Y)\exp(Z)=\exp({AX+BZ+CY+DI}) \ , $$ derived in arXiv:1502.06589,
JHEP {\bf 1505} (2015) 113. This includes, as a particular case, $\exp(X)
\exp(Z)$, with $[X,Z]$ containing other elements in addition to $X$ and $Z$.
The algorithm exploits the associativity of the BCH formula and is based on the
decomposition $\exp(X)\exp(Y)\exp(Z)=\exp(X)\exp({\alpha Y}) \exp({(1-\alpha)
Y}) \exp(Z)$, with $\alpha$ fixed in such a way that it reduces to
$\exp({\tilde X})\exp({\tilde Y})$, with $\tilde X$ and $\tilde Y$ satisfying
the Van-Brunt and Visser condition $[\tilde X,\tilde Y]=\tilde u\tilde X+\tilde
v\tilde Y+\tilde cI$. It turns out that $e^\alpha$ satisfies, in the generic
case, an algebraic equation whose exponents depend on the parameters defining
the commutator algebra. In nine {\it types} of commutator algebras, such an
equation leads to rational solutions for $\alpha$. We find all the equations
that characterize the solution of the above decomposition problem by combining
it with the Jacobi identity.
