Classification of commutator algebras leading to the new type of closed Baker-Campbell-Hausdorff formulas

We show that there are 13 types of commutator algebras leading to the new closed forms of the Baker\u2013Campbell\u2013Hausdorff (BCH) formula exp(X)exp(Y)exp(Z)=exp(AX+BZ+CY+DI), Turn MathJax on derived in Matone (2015). 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(\u3b1Y)exp((1 12\u3b1)Y)exp(Z), with \u3b1 fixed in such a way that it reduces to View the MathML source, with View the MathML source and View the MathML source satisfying the Van-Brunt and Visser condition View the MathML source. It turns out that e\u3b1 satisfies, in the generic case, an algebraic equation whose exponents depend on the parameters defining the commutator algebra. In nine types of commutator algebras, such an equation leads to rational solutions for \u3b1. We find all the equations that characterize the solution of the above decomposition problem by combining it with the Jacobi identity