  We consider the $L_2$-regularity of solutions to backward stochastic
differential equations (BSDEs) with Lipschitz generators driven by a Brownian
motion and a Poisson random measure associated with a L\'{e}vy process
$(X_t)_{t\in[0,T]}$. The terminal condition may be a Borel function of finitely
many increments of the L\'{e}vy process which is not necessarily Lipschitz but
only satisfies a fractional smoothness condition. The results are obtained by
investigating how the special structure appearing in the chaos expansion of the
terminal condition is inherited by the solution to the BSDE.
