Sasha Premet has provided a concrete direct approach to the question, which at first sight looks convincing. But I'd like to follow up with more detail about one aspect of the question. First, the paper by Tits which he mentions is Normalisateurs de tores I, J. Algebra 4 (1966), 96-116 (alas there is no part II). Basically Tits is investigating the relationship between the normalizer of a maximal torus, in a split reductive group over some field, and the resulting Weyl group. But he doesn't quite state explicitly here in which simple groups the extension splits (in other words, when is the Weyl group naturally a subgroup of the algebraic group).
Later some topologists at Rice (Morton Curtis, Alan Wiederhold, Bruce Williams) wrote a couple of joint papers, the first one Normalizers of maximal tori appearing in Springer Lect. Notes in Math. 418 (1974), 31-47. Here their framework involves compact connected semisimple Lie groups, so a standard translation into the language of complex Lie algebras or corresponding algebraic groups is needed. But they do show directly that two such Lie groups are isomorphic if and only if the respective normalizers are isomorphic (Theorem 1). Along with this, they work out explicitly a table (Theorem 2) showing which compact simple Lie groups (simply connected or taken modulo center) have a natural subgroup isomorphic to the Weyl group. Here only type
$G_2$ among the exceptional groups has such a splitting of the normalizer. The method is reasonable but too long to outline here.
Amusingly, they write at first: It seems strange that Theorems 1 and 2 do not seem to have been known, because their proofs use no techniques not known for many years. But then an added footnote states that they learned recently that in unpublished work Tits had earlier proved Theorem 2 in a more comprehensive way.
One other comment is that automorphisms of finite order of semisimple complex Lie algebras play a major role in the work of Victor Kac and are discussed in his book on infinite dimensional Lie algebras as well as in section X.5 of the classic book by Helgason Differential Geometry, Lie Groups, and Symmetric Spaces. But these sources don't seem to address the specific question asked.