This question comes from Example 4.2 of the Gekhtman-Shapiro-Vainshtein book Cluster Algebras and Poisson Geometry which I have attached. My goal is to understand how to compute $\{x_1’,x_2’\}=-\lambda x_2^2/x_1-\mu x_2x_3/x_1-\lambda x_2x_3/x_1$ knowing already that $\{x_1,x_2\}=\lambda x_1x_2,\{x_2,x_3\}=\mu x_2x_3,$ and $\{x_1,x_3\}=\nu x_1x_3$ (by log-canonicality).
My thought was to use the Leibniz identity $\{f_1f_2,f_3\}=f_1\{f_2,f_3\}+\{f_1,f_3\}f_2$ by taking $f_1=1/x_1, f_2=x_2+x_3,$ and $f_3=x_2$ which gives $\{x_1’,x_2’\}=\{x_2+x_3,x_2\}/x_1+\{1/x_1,x_2\}(x_2+x_3).$ That first bracket term will give me the $-\mu x_2x_3$ but for the second bracket term I do not know how to compute $\{1/x_1,x_2\}$ from knowing $\{x_1,x_2\}.$
Any guidance is appreciated as I am very new to Lie brackets and Poisson brackets.
In local coordinates $(x^1, \ldots, x^d)$ you can always calculate the Poisson bracket of arbitrary functions in terms of Poisson brackets of the coordinates $\{x^i, x^j\}$ and first derivatives of the functions (by the fact that we have a skew-symmetric bi-derivation, combined with Taylor's theorem, see e.g. [Lectures on Poisson Geometry, Proposition 14]): $$\{f,g\} = \sum_{i,j=1}^d \{x^i, x^j\} \frac{\partial f}{\partial x^i} \frac{\partial g}{\partial x^j} = \sum_{1 \leq i < j \leq d} \{x^i, x^j\} \bigg(\frac{\partial f}{\partial x^i} \frac{\partial g}{\partial x^j} - \frac{\partial f}{\partial x^j} \frac{\partial g}{\partial x^i}\bigg)$$
In particular in coordinates $(x_1,x_2,x_3)$ we have $\{1/x_1, x_2\} = \{x_1,x_2\}\frac{\partial}{\partial x_1}\big(1/x_1\big)\frac{\partial x_2}{\partial x_2} = -\{x_1,x_2\}/x_1^2$.