Let $\mathfrak{g}$ be a dimension 3 Lie algebra and $[\quad,\quad]$ be a rank 1 map from $\bigwedge^{2}\mathfrak{g} \rightarrow \mathfrak{g}$. In this case, the kernel of $[\quad,\quad]$ is $3 - 1 = 2$ dimensional. Why does this mean that for some $X \in \mathfrak{g}$, the kernel consists of all vectors of the form $X \wedge Y$ with $Y$ ranging over all of $\mathfrak{g}$?
Kernel of the Lie bracket
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linear-algebra
lie-algebras