Regular elements form a connected dense open subset

In any fixed basis of $\mathfrak{g}$, the matrix entries of ${\mathrm{ad}}_x$ depend linearly on $x$. By Regular elements and rank for a complex semisimple Lie algebra, regularity is the condition that $\mathrm{rank}({\mathrm{ad}}_x)=n-r$, so the nonregular locus is cut out by the vanishing of all $(n-r)\times (n-r)$ minors and is therefore Zariski closed.

Standard structure theory for complex semisimple Lie algebras supplies at least one regular element, so the complement of the closed set from step 1.1 is a nonempty Zariski-open subset. Because a nonempty Zariski-open subset of the complex affine space underlying $\mathfrak{g}$ is dense, the regular set is dense.

The complement of a proper complex algebraic subset of a finite-dimensional complex vector space is connected, so the nonempty open regular set from step 2.1 is connected as well. Hence the regular elements form a connected dense open subset.