Lie's theorem

If $\mathfrak{g}=0$, any nonzero $w\in V$ is a common eigenvector with the zero functional.

Assume $\mathfrak{g}\mathrm{\ne }0$ and the theorem for solvable Lie algebras of smaller dimension.

By [L1], choose a codimension-one ideal $\mathfrak{h}$ and write $\mathfrak{g}=\mathfrak{h}\oplus kx$. Its derived series is contained termwise in that of $\mathfrak{g}$, so $\mathfrak{h}$ is solvable. Step 1.2 gives $0\mathrm{\ne }v\in V$ and $\lambda \in {\mathfrak{h}}^{\ast }$ with $hv=\lambda (h)v$ for every $h\in \mathfrak{h}$.

Put $W_j=\mathrm{span}(v,xv,\dots ,x^{j}v)$ and $W={\sum }_{j\ge 0}{W}_j$; finite dimensionality makes $W$ finite-dimensional and $x$-stable. Induction on $j$, using $h{x}^j=x^{j}h+{\sum }_{i=0}^{j-1}{x}^i[h,x]x^{j-1-i}$ and $[h,x]\in \mathfrak{h}$, shows $h{W}_j\subseteq W_j$ and $h{x}^{j}v\equiv \lambda (h)x^{j}v(\mathrm{m}\mathrm{o}\mathrm{d}{W}_{j-1})$. Thus $W$ is $\mathfrak{g}$-stable and every $h\in \mathfrak{h}$ acts upper triangularly on a basis extracted from the cyclic list, with constant diagonal $\lambda (h)$.

Let $d=\dim W>0$. Because $W$ is stable, [L2] and [L4] give $0={\mathrm{tr}}_W([x,h])=d\lambda ([x,h])$ for every $h\in \mathfrak{h}$, where the last equality uses the constant diagonal from step 3.1. Characteristic zero makes $d\cdot 1_k\mathrm{\ne }0$, so $\lambda ([x,h])=0$. This is the exact use of the characteristic hypothesis.

The nonzero common $\mathfrak{h}$-weight space $V_{\lambda }=\{u:hu=\lambda (h)u\text{ for all }h\in \mathfrak{h}\}$ contains $v$. It is $x$-stable: for $u\in V_{\lambda }$, [L2] and step 4.1 give $h(xu)=x(hu)+[h,x]u=\lambda (h)xu+\lambda ([h,x])u=\lambda (h)xu$.

By algebraic closure and [L3], $x{\mathrm{\mid }}_{V_{\lambda }}$ has a nonzero eigenvector $w$, say $xw=aw$. Then $hw=\lambda (h)w$ for $h\in \mathfrak{h}$, and for $y=h+tx$ we have $yw=(\lambda (h)+ta)w$. This defines the required linear functional $\chi$. Algebraic closure is used only in [L3], characteristic zero only in step 4.1, and all choices are a finite sequence of existential choices rather than AC.