Pensieve header: Better solving the yaxyax equations.

Representing (𝔤^ϵ).

ME=MatrixExp; MF=MatrixForm;
Simp[sol_] := Flatten[sol]/. ConditionalExpression[ℰ_,_]:> ℰ/. (var_->val_) :> (var -> Simplify[PowerExpand@val]);

ρt=(({{1, 0}, {0, 1}}));  ρy=(({{0, 0}, {-ϵ, 0}}));  ρa=(({{(1+1/ϵ)/2, 0}, {0, -(1-1/ϵ)/2}})); ρx=(({{0, 1}, {0, 0}}));
Simplify@{ρa.ρx-ρx.ρa==ρx, ρa.ρy-ρy.ρa==-ρy, ρx.ρy-ρy.ρx==ρt-2ϵ ρa}

Simplify[ρt-2ϵ ρa] // MF

MF /@ {MatrixFunction[(T^(#))&, ρt], MatrixFunction[(A^(#))&, ρa]}

eqn=Simplify@PowerExpand[MatrixFunction[(T1^(#))&, ρt].ME[η1 ρy].MatrixFunction[(A1^(#))&, ρa].ME[ξ1 ρx].MatrixFunction[(T2^(#))&, ρt].ME[η2 ρy].MatrixFunction[(A2^(#))&, ρa].ME[ξ2 ρx]==MatrixFunction[(T0^(#))&, ρt].ME[η0 ρy].MatrixFunction[(A0^(#))&, ρa].ME[ξ0 ρx]];
MF /@ eqn

sol=Solve[Thread[Flatten/@eqn], {T0,η0,A0,ξ0}, Reals]

MF /@ {MatrixFunction[(T^(#))&, ρt/ϵ], MatrixFunction[(A^(#))&, ρa]}

eqn=Simplify@PowerExpand[MatrixFunction[(T1^(#))&, ρt/ϵ].ME[η1 ρy].MatrixFunction[(A1^(#))&, ρa].ME[ξ1 ρx].MatrixFunction[(T2^(#))&, ρt/ϵ].ME[η2 ρy].MatrixFunction[(A2^(#))&, ρa].ME[ξ2 ρx]==MatrixFunction[(T0^(#))&, ρt/ϵ].ME[η0 ρy].MatrixFunction[(A0^(#))&, ρa].ME[ξ0 ρx]];
MF /@ eqn

sol=Solve[Thread[Flatten/@eqn], {T0,η0,A0,ξ0}]

eqn=Simplify[ME[η1 ρy].ME[α1 ρa].ME[ξ1 ρx].ME[η2 ρy].ME[α2 ρa].ME[ξ2 ρx]==T0*ME[η0 ρy].ME[α0 ρa].ME[ξ0 ρx]];
MF /@ eqn

sol=Solve[Thread[Flatten/@eqn], {T0,η0,α0,ξ0}, Reals]

Simp[sol]

FullSimplify[(E^(-(((α1+α2) (-1+ϵ))/(2 ϵ)))) (1-(E^(α1)) ϵ η1 ξ1-(E^(α2)) ϵ η2 ξ2+(E^(α1+α2)) ϵ η1 (-1+ϵ η2 ξ1) ξ2)]

ME[η1 ρy].ME[α1 ρa].ME[ξ1 ρx].ME[η2 ρy].ME[α2 ρa].ME[ξ2 ρx] // LUDecomposition // Simplify //First// MF

Simplify /@ Det /@ eqn

sol=Solve[{Simplify /@ Det /@ eqn}~Join~Thread[Flatten/@eqn], {τ0,η0,α0,ξ0}]

M=Table[Which[i>j,0,i==j,1,i<j,(Subscript[a, i,j])], {i,5}, {j,5}];
M // MF

MatrixLog[M] // Simplify//MF

M=With[{n=4},Table[(Subscript[a, i,j]), {i,n}, {j,n}]];
M // MF

M // LUDecomposition // First // Simplify // MF