a constant Poisson structure $\alpha(x) = \alpha$. The main Theorem tells us that the generating function is $S_h(p_1,p_2,x)= S_0(p_1,p_2,x) + h p_1^t\alpha p_2$. The multiplication space can then be described as $$G^{(m)}_h=\Big\{\Big((p_1,x+h\alpha p_2),(p_2,x-h\alpha p_1),(p_1+p_2,x)\Big),(p_1,p_2,x)\in B_2\Big\}.$$ By Proposition \[formalgroupoid\] the structure maps are given by

$$\begin{aligned}
\epsilon_h(x) & = & (0,x)\\
i_h(p,x) & = & (-p,x)\\
s_h(p,x) &=& x-h \alpha p\\
t_h(p,x)& =& x+h\alpha p.\end{aligned}$$

### Linear Poisson structure

Suppose that we have a linear Poisson structure $\alpha^{ij}(x)=\alpha^{ij}_kx^k$ on $\mathbb R^d$ which can then be considered as the dual of a Lie algebra $\mathbb R^d = \mathcal G^*$, the bracket on $\mathcal G$ being given by $[\epsilon^i,\epsilon^j] = 2\alpha^{ij}_k \epsilon^k$, where $\epsilon^l$, $l=1,\dots,d$ is a basis of $\mathcal G (= \mathbb R^{*d})$. For this Lie algebra we have the CBH formula $\exp(p_1)\exp(p_2)=\exp(CBH(p_1,p_2))$, $$CBH(p_1,p_2)= p_1+p_2+\frac{1}{2}[p_1,p_2]
+\frac{1}{12}\Big([p_1,[p_2,p_2]]+[p_2,[p_2,p_1]]\Big)+\dots,$$

It is easy to check directly that $$S_h(p_1,p_2,x) = \langle\frac{1}{h}CBH(h p_1,h p_2),x\rangle$$ where $\langle.,.\rangle$ is the usual pairing between $\mathcal G$ and $ \mathcal G^*$, satisfies the SGA equation. It is equivalent to the associativity of CBH, i.e., $$CBH(hp_1,CBH(hp_2,hp_3)) = CBH(CBH(hp_1,hp_2),hp_3).$$

By the uniqueness of the generating function given by the main Theorem we recover a result of V. Kathotia (see [@kathotia1998]):

For the Poissson structure 