Understanding the regular element of a Lie algebra

Question

Given a Lie algebra $\mathfrak{g}$, we defined $$ \mathfrak{g}_{0,x}=\{ y\in \mathfrak{g}: \exists N>0\ ad(x)^N(y)=0\}, $$ and $x$ is said to be regular if $\mathfrak{g}_{0,x}$ is of minimal dimension. So I am kind of confused here what the minimal mean. Does it mean $\mathfrak{g}_{0,x}$ has the smallest dimension among such $\mathfrak{g}_{0,x}$ for all $x\in \mathfrak{g}$? Or does it mean $\mathfrak{g}_{0,x}$ has the smallest dimension among all subalgebras of $\mathfrak{g}$?

Answer 1

It means the first. It cannot mean the last because the smallest subalgebra of any Lie algebra is $0$, and convince yourself that for every $x\neq 0$, the dimension of $\mathfrak g_{0,x}$ is $\ge 1$ (usually much bigger).