I'm reading the book: Coherent State in Quantum Physics, by Jean-Pierre Gazeau. In the page no. 35 of this book, the Weyl-Heisenberg Lie Algebra $\mathfrak{w}_m$ has been given that $$\mathfrak{w}_m=linear\,span\{iQ,iP,iI_d\};$$ where $Q=\frac{a+a^\dagger}{\sqrt{2}}$, $P=\frac{a-a^\dagger}{i\sqrt{2}}$ and $I_d$ is the identity operator with $a$ is annihilation operator and $a^\dagger$ is creation operator. One can see that a generic element of $\mathfrak{w}_m$ is written as $$\mathfrak{w}_m\ni X=isI_d+i(pQ-qP)=isI_d+(za^\dagger-\overline{z}a);$$ where $s\in\mathbb{R}$ and $z=\frac{q+ip}{\sqrt{2}}$. Here $X$ is the anti-self-adjoint and the infinitesimal generator of the unitary operator: $$e^X = e^{is}e^{za^\dagger-\overline{z}a}:=e^{is}D(z).$$ After this some discussion on the action of the D-Function. My question is, in the page no. 37, it is said that the map $X\longmapsto e^X=e^{is}D(z)$ is irreducible unitary representation of Weyl-Heisenberg Algebra, but first of all it, to say this map $X\longmapsto e^X=e^{is}D(z)$ is representation of Weyl-Heisenberg Algebra, I think it should satisfy the relation $$e^{[X,Y]}=[e^X,e^Y],$$ for all $X,Y \in \mathfrak{w}_m$, since it is Lie algebra. But I'm unable to obtain this relation. Could anyone please help me...!!! I'm looking for a explanation. Thank you in advance for any help.
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