Let $G$ be a Lie group and $g$ the associated Lie algebra. Let $X,Y\in g$. Is there a formula in term of $a=\exp(tX)$ and $b=\exp(tY)$ for $\exp(t[X,Y])$? (Let $H$ be the group generated by $\exp tX$ and $\exp tY$. Let $h$ be the Lie algebra of $H$. Then $h$ contains $X$ and $Y$. Since $[X,Y]\in h$, $\exp(t[X,Y])\in H$, hence I expect that there is such a formula). By a formula I mean an expression:
$$\exp(t[X,Y])=\exp(t_1X)\exp(s_1Y)\dots\exp(t_kX)\exp(s_kY)$$
where $t_i$, $s_i$ and $k$ are functions of $t,X,Y$.
The Baker-Campbell-Hausdorff series has $1$ product and infinitely many brackets. The Zassenhaus series has infinitely many products and brackets. Is there a similar formula with few brackets and infinitely many products (the above expression would be one with $1$ bracket)?
It's not BCH (express group structure in terms of the Lie algebra laws), but rather its inversion (express the Lie algebra laws, here the bracket, in terms of the group law); it's much harder and was addressed/done by Lazard:
Michel Lazard. Sur les groupes nilpotents et les anneaux de Lie (French) Annales scientifiques de l’É.N.S. 3e série, tome 71, no 2 (1954), p. 101-190. Freely available at Numdam
See notably (2.8) p157.