Let $\mathbb{k}$ be a field, $\mathfrak{g}$ be a finite-dimensional Lie algebra over $\mathbb{k}$. In Bourbaki's "Lie Groups and Lie Algebras", Ch I, he defines four radical-like ideals of $\mathfrak{g}$:

- the radical $\mathfrak{r}$, i.e. the maximal solvable ideal;
- the radical of Killing form $\mathfrak{k}$, i.e. $\mathfrak{k}=\mathfrak{g}^\perp$;
- the maximal nilpotent ideal $\mathfrak{n}$;
- the nilpotent radical $\mathfrak{s}$, i.e. intersection of all kernels of irreducible finite-dimensional representations of $\mathfrak{g}$.

He also shows that, $\mathfrak{r}=[\mathfrak{g},\mathfrak{g}]^\perp$, $\mathfrak{s}=[\mathfrak{g}, \mathfrak{g}]\cap\mathfrak{r}=[\mathfrak{g},\mathfrak{r}]$, and the following inclusion relations: $$\mathfrak{r} \supset \mathfrak{k} \supset \mathfrak{n} \supset \mathfrak{s}.$$

On the other hand, in Jacobson's book, the nilpotent radical is defined as $\mathfrak{n}$.

My question is, does $\mathfrak{s}$ coincide with $\mathfrak{n}$? or, are Bourbaki's nilradical and Jacobson's nilradical equivalent?

I guess the answer is NO (otherwise Bourbaki should have proved it), but I cannot find any example. Could any one give me an example? Thank you!

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