Pensieve header: The yaxyax formulas for vanilla sl(2).

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

ρt=(({{1, 0}, {0, 1}}));  ρy=(({{0, 0}, {1, 0}}));  ρa=(({{1, 0}, {0, -1}})); ρx=(({{0, 1}, {0, 0}}));

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

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

Simp[sol]

Simplify[eqn /. Simp[sol]]