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 idea 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 realizations in the mathematical physics literature (and are direct extensions of the corresponding descriptions of the undeformed cases, where UEA of Lie (super)algebras are shown to be isomorphic to undeformed parabosonic or parafermionic algebras). 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 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.
Latter on, on the same paper (see p. 2611, Proposition 4), a description of the hopf subalgebra
$$
U_q\big(gl(n)\big)\subset U_q\big(osp(1/2n)\big)
$$
in terms of deformed parabosonic generators is provided. The $U_q\big(gl(n)\big)$ hopf subalgebra is described in terms of a suitable subset of deformed parabosonic generators and relations.
Similar methods can be used to describe 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.