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In Carter's Lie Algebras of Finite and Infinite Type he uses the following fact

We recall from Lemma 2.1 that the 1-dimensional representations of L are in bijective correspondence with linear maps $L/L^2 \to \mathbb C$. The vector space $L/L^2$ over $\mathbb C$ cannot be expressed as the union of finitely many proper subspaces.

in the proof of Theorem 2.9:

Theorem 2.9 Let $L$ be a nilpotent Lie algebra and $V$ a finite dimensional $L$-module. For any 1-dimensional representation $\lambda$ of $L$ we define $V_\lambda=\{v \in V$ | for each $x \in L$ there exists $N(x)$ such that $\left.(\rho(x)-\lambda(x) 1)^{N(x)} v=0\right\}$. Then $$ V=\bigoplus_\lambda V_\lambda $$ and each $V_\lambda$ is a submodule of $L$.

Why is this fact true? I would have though that since $L$ is finite dimensional and so is $L^2$ then the quotient is also and then simply taking some basis gives the decomposition into proper subspaces.

JvT
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    He does say union and no vector space (over an infinite field) can be expressed as the union of finitely many proper subspaces. – Callum Feb 04 '25 at 07:53
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    That's a standard linear algebra fact, see e.g. https://math.stackexchange.com/q/145869/96384, https://math.stackexchange.com/q/1165705/96384, maybe most generic and general version in https://math.stackexchange.com/q/60698/96384. – Torsten Schoeneberg Feb 04 '25 at 21:57
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    Thanks for pointing that out. I was mistakenly thinking about direct sums, not unions. – JvT Feb 05 '25 at 07:41

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