Since the OP is asking for examples of
sub-Hopf algebras which are not generated by the standard generators
i.e. the Chevalley generators (which are actually the generators of the Cartan-weyl basis with a different normalization), satisfying the Chevalley-Serre Relations, maybe the following method might appear useful for constructing such examples:
The quantum universal enveloping (super)algebras $U_q(g)$, where $g$ is a simple Lie (super)algebra, are known to have alternative -isomorphic- descriptions in terms of $q$-deformed paraparticle (parabosonic or parafermionic) algebras: such descriptions are called pre-oscillator or paraparticle realizations in the mathematical physics literature.
(They extend the corresponding oscillator realizations of the undeformed cases, where UEA of Lie (super)algebras are shown to be isomorphic or homomorphic to the usual (undeformed) parabosonic or parafermionic algebras or homomorphic to Weyl and Clifford algebras i.e. the CCR and the CAR relations of quantum mechanics).
See for example at this article for $U_q\big(osp(1/2n) \big)$ written in terms of $q$-deformed parabosonic generators or this article for the case of $U_q\big(so(2n+1) \big)$ written in terms of $q$-deformed parafermionic generators.
In the paper: The quantum superalgebra $U_q\big( osp(1/2n)\big)$: deformed parabose operators and root of unity representations, the authors use deformed realizations to connect the rep theory of $U_q\big( osp(1/2n)\big)$ with the deformed paraboson Fock spaces. They describe the isomorphism between the $q$-deformed UEA of the Lie superalgebra $osp(1/2n)$ and the $q$-deformed parabosonic algebra: see p. 2608, relations (2.12) and their converse: (2.13), (2.14), (2.15) for the undeformed case and p. 2609, relation (3.3), (3.4) and their converse in p. 2610, rel. (3.5), for the deformed case.
These realizations, are then used (see p. 2611, Proposition 4), to provide a description of the sub-hopf algebra
$$
U_q\big(gl(n)\big)\subset U_q\big(osp(1/2n)\big)
$$
in terms of deformed parabosonic generators. The $U_q\big(gl(n)\big)$ sub-hopf algebra is described in terms of a suitable subset of deformed parabosonic generators and relations (which are not linear combinations of the Cartan-Weyl generators).
Similar methods, utilizing bosonic, fermionic or paraparticle generators, can be used to describe -and maybe used as a tool at some classification attempt- other hopf subalgebras of various $q$-deformed UEAs. There are lots of works in a similar spirit in the mathematical physics literature - see also the references in the cited articles.
I hope the above might be of some interest for the purposes of the OP.