Noetherianity up to conjugation of locally diagonal inverse limits

We prove that the inverse limit of the sequence dual to a sequence of Lie algebras is Noetherian up to the action of the direct limit of the corresponding sequence of classical algebraic groups when the sequence of groups consists of diagonal embeddings. We also classify all conjugation-stable closed subsets of the space of $\mathbb{N}\times\mathbb{N}$ matrices.