On multivariable Zassenhaus formula

We give a recursive algorithm to compute the multivariable Zassenhaus formula {\mathrm{e}}^{{{\displaystyle X}}_{\mathrm{1}}+{{\displaystyle X}}_{\mathrm{2}}+\mathrm{...}+{{\displaystyle X}}_n}={\mathrm{e}}^{{{\displaystyle X}}_{\mathrm{1}}}{\mathrm{e}}^{{{\displaystyle X}}_{\mathrm{2}}}\mathrm{...}{\mathrm{e}}^{{{\displaystyle X}}_n}{{\displaystyle \mathrm{\Pi }}}_{k=\mathrm{2}}^{\infty }{\hspace{0.17em}\mathrm{e}}^{{{\displaystyle W}}_k} and derive ane effective recursion formula of {{\displaystyle W}}_k.