Let $\mathfrak{g}$ be a Lie algebra over algebraically closed field $k$ of characteristic $0$.

The radical $R(\mathfrak{g})$ is the largest solvable ideal of $\mathfrak{g}$.

The Killing form $\kappa$ is the trace form of the adjoint representation of $\mathfrak{g}$.

The radical of the Killing form on $\mathfrak{g}$ is $\mathfrak{g}^\perp = \{x \in \mathfrak{g} : \kappa(x,y) = 0, \forall y \in \mathfrak{g} \}.$

Could anyone help me find a concrete example of a Lie algebra $\mathfrak{g}$ and and an element $x \in \mathfrak{g}$ such that $x \in R(\mathfrak{g})$ but $x \not\in \mathfrak{g}^\perp$?


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