# Proposition 4.5 of Brian Hall - Lie Groups, Lie Algebras, and Representations

I have a question about the proof of Proposition 4.5 in Brian Hall's Lie Groups, Lie Algebras, and Representations.

The proposition says that

Let $$G$$ be a connected matrix Lie group with Lie algebra $$\mathfrak{g}$$. Let $$\Pi$$ be a representation of $$G$$ and $$\pi$$ the associated representation of $$\mathfrak{g}$$. Then $$\Pi$$ is irreducible iff $$\pi$$ is irreducible.

Overall I understand the proof besides one part in the proof of the backward statement. The proof says

Let $$W$$ be an invariant subspace for $$\Pi$$. Then $$W$$ is invariant under $$\Pi(e^{tX})$$ for all $$X \in \mathfrak{g}$$ and hence under $$\pi(X) = \frac{d}{dt} \Pi(e^{tX}) |_{t=0}$$.

Why does $$W$$ being invariant under $$\Pi(e^{tX})$$ imply it is invariant under $$\frac{d}{dt} \Pi(e^{tX}) |_{t=0}$$?

Here's the screenshot of the proposition and its proof.

This is just calculus. Let $$L_t$$ be a family of linear operators that send a subspace $$V$$ to $$V$$, then $$\frac{L_t(x)-L_0(x)}{t}\in V$$ for any $$x\in V, t\not=0$$. And by the completeness of $$V$$, $$\lim_{t\rightarrow 0}\frac{L_t(x)-L_0(x)}{t}\in V$$.