§ Lie bracket commutator from exponentiation (TODO)

I thought this was quite cute. To make sense of [a,b][a, b] we consider the expression eϵaeϵbeϵaeϵbe^{\epsilon a} e^{\epsilon b} e^{- \epsilon a} e^{- \epsilon b} which is equivalent to:
(1+ϵa+ϵ2a2/2)(1+ϵb+ϵ2b2/2)(1ϵa+ϵ2a2/2)(1ϵb+ϵ2b2/2)[(1+ϵa+ϵ2a2/2)(1ϵa+ϵ2a2/2)][(1+ϵb+ϵ2b2/2)(1ϵb+ϵ2b2/2)][((1+ϵ2a2/2)+ϵa)((1+ϵ2a2/2)ϵa)][(1+ϵ2b2/2+)ϵb)((1+ϵ2b2/2)ϵb)][((1+ϵ2a2/2)2ϵ2a2)][(1+ϵ2b2/22ϵ2b2)] \begin{aligned} &(1 + \epsilon a + \epsilon^2 a^2/2)(1 + \epsilon b+ \epsilon^2 b^2/2)(1 - \epsilon a + \epsilon^2 a^2/2)(1 - \epsilon b + \epsilon^2 b^2/2) \\ &[(1 + \epsilon a + \epsilon^2 a^2/2)(1 - \epsilon a + \epsilon^2 a^2/2)] [(1 + \epsilon b+ \epsilon^2 b^2/2)(1 - \epsilon b + \epsilon^2 b^2/2)] \\ &[((1 +\epsilon^2 a^2/2) + \epsilon a)((1 + \epsilon^2 a^2/2) - \epsilon a)] [(1 + \epsilon^2 b^2/2+ ) \epsilon b) ((1 + \epsilon^2 b^2/2) - \epsilon b)] \\ &[((1 +\epsilon^2 a^2/2)^2 - \epsilon^2 a^2)] [(1 + \epsilon^2 b^2/2^2 - \epsilon^2 b^2)] \\ \end{aligned}
TODO