Formula (12) in the paper
expresses the derivative of an exponential of a parameter-dependent Lie algebra element. The authors call it 'Duhamel's formula', without giving a reference.
Where and in which context did Duhamel prove this formula? (I know the formula as Kubo's formula, from his work in statistical mechanics.)
Duhamel's formula for the derivative of the exponent of a matrix refers to Jean-Marie-Constant Duhamel, who described it in Eléments de calcul infinitésimal (volume 2, 1856; page 36) as a method to obtain solutions to inhomogeneous linear differential equations in terms of the solution to the homogenous equation.
In the context of Lie groups the formula for the derivative of the exponential map is attributed to Schur (1891).
A modern proof is in these notes. or alternatively here (theorem 3). A formal proof is given by \begin{align} \frac{d}{dt}e^{A(t)} &= \lim_{N \to \infty}\frac{d}{dt}\left(1 + \frac{A(t)}{N}\right)^N\\ &= \frac{1}{N}\lim_{N \to \infty}\sum_{k=1}^N\left(1 + \frac{A(t)}{N}\right)^{k-1}A'(t)\left(1 + \frac{A(t)}{N}\right)^{N-k}~\\ &=\int_0^1 e^{sA(t)}A'(t)e^{(1-s)A(t)}\,ds. \end{align}
