Let $\mathcal{L}$ be a quasicompact cocycle on a separable Banach space $X$ with its Borel $\sigma$-algebra, with top exponent $\lambda_1 \in \mathbb{R}$ of multiplicity $m_1$ and second exponent $\lambda_2$ (possibly $-\infty$), almost surely. Then there are

• a fast space $E_1$ of multiplicity $m_1$ at rate $\lambda_1$, with eventually positive growth;
• a measurable family of closed complements $V(\omega)$ of $E_1(\omega)$ of codimension $m_1$, with $\mathcal{L}_\omega V(\omega) \subseteq V(\sigma\omega)$;
• strongly measurable, tempered idempotents $P_\omega$ with range $V(\omega)$ and kernel containing $E_1(\omega)$,

such that $\frac1n \log \|\mathcal{L}^{(n)}_\omega P_\omega\| \to \lambda_2$ almost surely.

This is Lemma 29 of Lee (2024).