Consider the differential equation (Schrödinger, but rewritten to be pleasing to Lie algebraic eyes):
$\frac{d U(t)}{dt} = c(t)U(t)$
where $c(t)=a+w(t)b(t)$, $a,b \in \mathfrak{su}(n)$ and $w$ is a smooth real valued function of $t$.
Assuming that $U(t)$ really does solve the equation given, I'm trying to find a general formula for:
$\frac{d^n U(t)}{dt^n} \bigg|_{t=0}$
in terms of $a,b$ and the derivatives of $w$ at $t=0$ only.
The first few are easy to find, but I'm really hoping that someone can spot the pattern and get a general formula as I'm not seeing it after substantial staring!