  In recent years, researchers have discovered various large algebraic
structures that have surprising finiteness properties, such as FI-modules and
Delta-modules. In this paper, we add another example to the growing list: we
show that certain degree two twisted commutative algebras are noetherian. This
example appears to have some fundamental differences from previous examples,
and is therefore especially interesting. Reflective of this, our proof
introduces new methods for establishing noetherianity that are likely to be
applicable in other situations. The algebras considered in this paper are
closely related to the stable representation theory of classical groups, which
is one source of motivation for their study.
