# On multi-variable Zassenhaus formula

## TL;DR

This paper introduces a recursive algorithm for computing the multivariable Zassenhaus formula, enabling efficient calculation of exponential decompositions in Lie algebra contexts.

## Contribution

It presents a novel recursive method and an effective recursion formula for the multivariable Zassenhaus expansion.

## Key findings

- Developed a recursive algorithm for multivariable Zassenhaus formula
- Derived an effective recursion formula for the terms
- Facilitates efficient computation of exponential decompositions

## Abstract

In this paper, we give a recursive algorithm to compute the multivariable Zassenhaus formula $$e^{X_1+X_2+\cdots +X_n}=e^{X_1}e^{X_2}\cdots e^{X_n}\prod_{k=2}^{\infty}e^{W_k}$$ and derive an effective recursion formula of $W_k$.