I would like to know if there exists a closed form of the Baker Campbell Hausdorff theorem subject to the conditions that $[x,[x,y]] \sim x$ and $[y,[x,y]] \sim y$.
The simple cases that I know a closed form exists are when $[x,y]$ is a scalar, then the expansion truncates as $\log(e^x e^y) = x + y + \frac{1}{2}[x,y]$ and if $[x,y] = sy$ for some constant $s$, then we have $\log(e^x e^y) = x + \frac{sy}{1-e^{-s}}$.
These lead me to believe that there should be some closed form since I know the higher order terms in the BCH expansion are proportional to $x$ and $y$.
My specific problem is trying to apply these with $x = a \frac{d^2}{dp^2} $ and $y = b p^2$ for some (possibly complex) constants $a$ and $b$. The commutator is $[x,y] = ab(4 p \frac{d}{dp} + 2)$ The nested commutators are $[x,[x,y]] = 8a^2b\frac{d^2}{dp^2}$ and $[y,[x,y]] = -8ab^2p^2 $.
Indeed these are proportional to the original x and y. Is there a known closed form of the BCH theorem for this example? Thanks in advance!