Pensieve header: Reproducing commutation relations from a representation; the classical case. (Wrong; aborted).

Code borrowed from pensieve://2017-06/MultiYax.nb.

Representing yax

ρ1=(({{0, 0, 1}, {0, 0, 0}, {0, 0, 0}}));ρy=(({{0, 0, 0}, {0, 0, ϵ}, {0, 0, 0}})); ρa=(({{0, 0, 0}, {0, -γ, 0}, {0, 0, 0}})); ρx=(({{0, 1, 0}, {0, 0, 0}, {0, 0, 0}})); ρ0=(({{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}));
B[x_?MatrixQ, y_?MatrixQ] := x.y-y.x;

{B[ρa,ρx]==γ ρx, B[ρa,ρy]==-γ ρy, B[ρx,ρy]==ϵ ρ1, B[ρ1,ρy]==ρ0, B[ρ1,ρa]==ρ0, B[ρ1,ρx]==ρ0}

The 2-Stitch of Exponentials

P=IdentityMatrix[3];
Do[
P=Expand[P.MatrixExp[(Subscript[η, i])ρy].MatrixExp[(Subscript[α, i])ρa].MatrixExp[(Subscript[ξ, i])ρx]],
{i,1,2}
];
P // MatrixForm

Column[
ϕ=First@Solve[Thread[Flatten/@(P==MatrixExp[η0 ρy].MatrixExp[α0 ρa].MatrixExp[ξ0 ρx].MatrixExp[τ0 ρ1])], {η0,α0,ξ0,τ0}] /. (v_->ℰ_) :> (v->Expand@PowerExpand[ℰ])
]

Φ=Simplify[t0 τ0 + y0 η0 + a0 α0 + x0 ξ0 /. ϕ]

Collect[Limit[Φ, ϵ->0], {t0, y0, a0,xξ0}]

Collect[Limit[Φ, ϵ->0], {t0, y0, a0,xξ0}]

Reproducing the Commutation Relations

Simplify@PowerExpand[((∂_(α1,ξ2))(E^Φ)-(∂_(ξ1,α2))(E^Φ)) /. {τ1|η1|α1|ξ1|τ2|η2|α2|ξ2 -> 0}]

Simplify@PowerExpand[((∂_(α1,η2))(E^Φ)-(∂_(η1,α2))(E^Φ)) /. {τ1|η1|α1|ξ1|τ2|η2|α2|ξ2 -> 0}]

qbxy=Simplify@PowerExpand[((∂_(ξ1,η2))(E^Φ)-q(∂_(η1,ξ2))(E^Φ)) /. {τ1|η1|α1|ξ1|τ2|η2|α2|ξ2 -> 0}]

Limit[qbxy, ϵ->0]