  We establish doubly-exponential degree bounds for Gr\"obner bases in certain
algebras of solvable type over a field (as introduced by Kandri-Rody and
Weispfenning). The class of algebras considered here includes commutative
polynomial rings, Weyl algebras, and universal enveloping algebras of
finite-dimensional Lie algebras. For the computation of these bounds, we adapt
a method due to Dub\'e based on a generalization of Stanley decompositions. Our
bounds yield doubly-exponential degree bounds for ideal membership and
syzygies, generalizing the classical results of Hermann and Seidenberg (in the
commutative case) and Grigoriev (in the case of Weyl algebras).
