Pensieve header: Products of multiple exponentials in the yax algebra; verifies Monoblog/170702.

Representing yax

ρ1=(({{0, 0, 1}, {0, 0, 0}, {0, 0, 0}}));ρy=(({{0, 0, 0}, {0, 0, τ}, {0, 0, 0}})); ρa=(({{0, 0, 0}, {0, -α, 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]==α ρx, B[ρa,ρy]==-α ρy, B[ρx,ρy]==τ ρ1, B[ρ1,ρy]==ρ0, B[ρ1,ρa]==ρ0, B[ρ1,ρx]==ρ0}

P=IdentityMatrix[3];
Do[
P=Expand[P.MatrixExp[(Subscript[η, i])ρy].MatrixExp[(Subscript[α, i])ρa].MatrixExp[(Subscript[ξ, i])ρx]],
{i,1,5}
];
P // MatrixForm

Column[
First@Solve[Thread[Flatten/@(P==MatrixExp[Y ρy].MatrixExp[A ρa].MatrixExp[X ρx].MatrixExp[σ ρ1])], {Y,A,X,σ}] /. (v_->ℰ_) :> (v->Expand@PowerExpand[ℰ])
]

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

Column[
First@Solve[Thread[Flatten/@(PS==MatrixExp[Y ρy].MatrixExp[A ρa].MatrixExp[X ρx].MatrixExp[σ ρ1])], {Y,A,X,σ}] /. (v_->ℰ_) :> (v->Expand@PowerExpand[ℰ])
]

Column[
First@Solve[Thread[Flatten/@(PS==MatrixExp[Y ρy+A ρa+X ρx+σ ρ1])], {Y,A,X,σ}] /. (v_->ℰ_) :> (v->Simplify@PowerExpand[ℰ])
]