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We show that there are {\it 13 types} of commutator algebras leading to thenew 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 thedecomposition $\exp(X)\exp(Y)\exp(Z)=\exp(X)\exp({\alpha Y}) \exp({(1-\alpha)Y}) \exp(Z)$, with $\alpha$ fixed in such a way that it reduces to$\exp({\tilde X})\exp({\tilde Y})$, with $\tilde X$ and $\tilde Y$ satisfyingthe Van-Brunt and Visser condition $[\tilde X,\tilde Y]=\tilde u\tilde X+\tildev\tilde Y+\tilde cI$. It turns out that $e^\alpha$ satisfies, in the genericcase, an algebraic equation whose exponents depend on the parameters definingthe commutator algebra. In nine {\it types} of commutator algebras, such anequation leads to rational solutions for $\alpha$. We find all the equationsthat characterize the solution of the above decomposition problem by combiningit with the Jacobi identity.

Classification of Commutator Algebras Leading to the New Type of Closed Baker-Campbell-Hausdorff Formulas