a=-(({{ρ, 0}, {0, σ}}))(α/(ρ-σ));b=-(({{σ, 0}, {0, ρ}}))(β/(ρ-σ));
ℏ=-(((ρ-σ))/(α β));
x=(({{0, 1}, {0, 0}}))(((E^(-ρ))-(E^(-σ)))/(-ℏ));y=(({{0, 0}, {1, 0}}));

(A=MatrixExp[-ℏ β a]; B=MatrixExp[-ℏ α b];)

{a.x-x.a==-α x, a.y-y.a==α y} // Simplify

{b.y-y.b==-β y, x.b-b.x==-β x} // Simplify

β a+α b // Simplify //MatrixForm

x.y-y.x==((B-A)/ℏ) // Simplify

MatrixExp[λ1 y].MatrixExp[λ2 (x.y-y.x)].MatrixExp[λ3 x] //Simplify// MatrixForm

eqn=Simplify[MatrixExp[μ x].MatrixExp[ν y]-MatrixExp[λ1 y].MatrixExp[λ2 (x.y-y.x)].MatrixExp[λ3 x]]

eqns=Flatten[eqn]

Series[eqns, {β,0,1}]

Series[eqns /. {λ1 -> ν+c β, λ2 -> μ ν, λ3 -> μ}, {β,0,1}]

c/.Solve[-c-(((E^(-ρ)) α μ (ν^2))/(ρ-σ))+(((E^(-σ)) α μ (ν^2))/(ρ-σ))==0, c] // Simplify

ser=Series[eqns /. {λ1 -> ν+(((E^(-ρ-σ)) ((E^ρ)-(E^σ)) α μ (ν^2))/(ρ-σ)) β+c1 (β^2), λ2 -> μ ν+c2 β, λ3 -> μ+c3 β}, {β,0,2}]

Solve[SeriesCoefficient[#,2]==0& /@ ser, {c1,c2,c3}]