This note investigates the symmetric composition \(e^{i\frac{1}{2}(W\omega ){\mathcal {X}}}e^{(Bt)\nabla }e^{i\frac{1}{2}(W\omega ){\mathcal {X}}}\) of the exponentiation of the Hamiltonian operator \(\nabla \) and the operator \({\mathcal {X}}=({\mathcal {X}}_1,\cdots ,{\mathcal {X}}_n)^\prime \) with \({\mathcal {X}}_j\) being multiplication by the j-th coordinate function for lattice translation matrix B and directional modulation matrix W. We point out that, in the case \(B=W=I\) , \(e^{i\frac{1}{2}\omega {\mathcal {X}}}e^{t\nabla }e^{i\frac{1}{2}\omega {\mathcal {X}}}\) gives rise to an alternative definition of the operator \(e^{t\nabla +i\omega {\mathcal {X}}}\) , which was originally defined by Folland from the approach of partial differential equation. We further extend the operator \(e^{t\nabla +i\omega {\mathcal {X}}}\) to anisotropic case \(e^{(Bt)\nabla +i(W\omega ){\mathcal {X}}}\) by setting \(e^{(Bt)\nabla +i(W\omega ){\mathcal {X}}}:=e^{i\frac{1}{2}(W\omega ){\mathcal {X}}}e^{(Bt)\nabla }e^{i\frac{1}{2}(W\omega ){\mathcal {X}}}\) and relate it to an initial problem of PDE. Finally, we focus on the operator \({\mathcal {R}}_r e^{i\frac{1}{2}(W\omega ){\mathcal {X}}}e^{(Bt)\nabla }e^{i\frac{1}{2}(W\omega ){\mathcal {X}}}{\mathcal {D}}_A\) with rotation \({\mathcal {R}}_r\) and nonsingular dilation \({\mathcal {D}}_A\) and study its coherent group structure by defining the so called ordered symplectic product \(\left[ (t,\omega ),({{\tilde{t}}}, {\tilde{\omega }})\right] ^A_{B,W}:=t^\prime \left( B^\prime (A^\prime )^{-1}W\right) {\tilde{\omega }}- {{\tilde{t}}}^\prime \left( B^\prime A^\prime W\right) \omega \) for the slice vectors \((t,\omega )\) and \(({\tilde{t}},\tilde{\omega })\) from the vectors \((r,t,\omega ,A)\) and \(({\tilde{r}},{\tilde{t}},\tilde{\omega },{\tilde{A}})\) in \({\mathbb {H}}:={\mathbb {R}}\times ({\mathbb {R}}^n\times {\mathbb {R}}^n)\times {\mathbb {M}}^{n\times n}\) , respectively.
