Pensieve header: Products of multiple exponentials in the β-yax algebra; continues pensieve://2017-06/BW.nb and pensieve://2017-07/.

ME=MatrixExp; MF=MatrixForm;

Representing β-yax

ℏ=((σ-ρ)/(γ β)); q=(E^(ℏ γ β));
y=(({{0, 0}, {-(E^ρ), 0}})); a=(γ/(ρ-σ))(({{ρ, 0}, {0, σ}}));x=(((E^(-σ))-(E^(-ρ)))/ℏ)(({{0, 1}, {0, 0}})); b=(β/(σ-ρ))(({{σ, 0}, {0, ρ}}));
MF /@ {A=ME[-ℏ β a], B=ME[-ℏ γ b], t=Simplify[β a-γ b] , T=ME[ℏ t]}

{a.x-x.a==γ x, x.A==q A.x, a.y-y.a==-γ y, b.y-y.b==-β y, x.y-q y.x==((({{1, 0}, {0, 1}}))-T.A.A)/ℏ} // Simplify

Clear[ℏ]; ρ=0; σ=ℏ β γ;
t=Simplify[β a-γ b]; A=ME[-β a/γ];  B=ME[-b];  T=ME[t/γ];
(#->MF@Simplify@ToExpression@#)& /@ {"{{q}}", "y","a","x","b", "t", "A", "B", "T"}

The 2-Stitch

P=(({{1, 0}, {0, 1}}));
Do[P=Expand[P.ME[(Subscript[η, i])y].ME[(Subscript[α, i])a].ME[(Subscript[ξ, i])x]],{i,2}];
Column[
sol=First@Solve[Thread[Flatten/@(P==τo ME[ηo y].ME[αo a].ME[ξo x])], {τo,ηo,αo,ξo}] /. (v_->ℰ_) :> (v->Simplify@PowerExpand[ℰ])
]

Column@Series[{ηo,αo,ξo,τo,((Log[τo])/(-β γ))} /.sol, {β,0,2}]

xy to yax

Column[
sol1=First@Solve[Thread[Flatten/@(ME[ξ x].ME[η y]==τo ME[ηo y].ME[αo a].ME[ξo x])], {τo,ηo,αo,ξo}] /. (v_->ℰ_) :> (v->FullSimplify@PowerExpand[ℰ])
]

Column@Series[{ηo,αo,ξo,τo,((Log[τo])/(-β γ))} /.sol1, {β,0,2}]