Pensieve header: An analysis of ${\mathfrak g}^\epsilon$ within $gl_2$.

ρw=(({{0, 1}, {0, 0}})); ρu=(({{0, 0}, {-ϵ, 0}}));  ρb=(({{-1, 0}, {0, -1}})); ρc=(({{-(1+1/ϵ)/2, 0}, {0, (1-1/ϵ)/2}})); ρ0=(({{0, 0}, {0, 0}}));
B[x_?MatrixQ, y_?MatrixQ] := x.y-y.x;
Simplify@{B[ρw,ρc]==ρw, B[ρc,ρu]==ρu, B[ρu,ρw]==ρb-2ϵ ρc}

ρb-ϵ ρc //Simplify // MatrixForm

MatrixForm/@Simplify /@ ((ρb-ϵ ρc)⊗ρc)

((MatrixForm/@Simplify /@ ((-2ρb+2ϵ ρc)⊗(-2ϵ ρc)))/(4ϵ))

MatrixExp[a ρw].MatrixExp[b ρu] // Simplify // MatrixForm

eqn=MatrixExp[a ρw].MatrixExp[b ρu]==MatrixExp[α ρu].MatrixExp[β ρw].MatrixExp[γ (ρb-2ϵ ρc)]

{sol}=Solve[Thread[Flatten/@eqn], {α,β,γ}]

sol=sol/.C[1]->0

Series[{α,β,γ}/.sol, {ϵ,0,1}]

Simplify[eqn/.sol]

Column@Series[{α,β,γ}/.sol, {ϵ,0,4}]