  For bounded domains $\Omega$ with Lipschitz boundary $\Gamma$, we investigate
boundary value problems for elliptic operators with variable coefficients of
fourth order subject to Wentzell (or dynamic) boundary conditions. Using form
methods, we begin by showing general results for an even wider class of
operators defined via two (intertwined) quadratic forms by defining very
abstract concepts of weak traces. Even in this general setting, we prove
generation of an analytic semigroup on the product space $L^2(\Omega) \times
L^2(\Gamma)$. Using recent results concerning weak co-normal traces, we apply
our abstract theory to the elliptic fourth-order case and are able to fully
characterize the domain in terms of Sobolev regularity, also obtaining
H\"older-regularity of solutions. Finally, we also discuss asymptotic behavior
and (eventual) positivity.
