Simple Lie Algebra Conjugacy and Dykin Diagrams

Question

Give two simple Lie Algebras $\mathfrak{g_1}$ and $\mathfrak{g_2}$, can we say anything about the conjugacy of $\mathfrak{g_1}$ and $\mathfrak{g_2}$ based on the properties of the corresponding Dynkin diagrams $\mathfrak{D_1}$ and $\mathfrak{D_2}$ ? (Say isomorphism or something ?)

Answer 1

Semisimple Lie algebra conjugacy (as matrix algebra conjugacy) is indeed equivalent to graph isomorphism. The article Lie algebra conjugacy by Joshua A. Grochow discusses this in detail, also for the general case of an arbitrary Lie algebra.