Conjugacy of Regular Semisimple Subalgebras

Question

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. In his paper Semisimple Subalgebras of Semisimple Lie Algebras, Dynkin states that by a result of Weyl,

1) Two regular semisimple subalgebras are conjugate if and only if their simple roots are conjugate under the Weyl group.

From this Dynkin concludes that for the classical Lie algebras,

2) Any regular semisimple subalgebra of a given type (e.g. $A_2\oplus B_2$ in $B_5$) is unique up to conjugacy, with some exceptions if $\mathfrak{g}$ is of type $D_n$. The same is true of the exceptional Lie algebras, except $E_7$ and $E_8$ have several exceptions.

Does anyone know where I can find a proof of Statements 1 and 2? The first statement supposedly follows from a result in Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen by Weyl, but I am unable to find an English translation. I am more interested in Statement 2, however, but Dynkin does not cite any source, nor does he hint at a proof.

Answer 1

It would help to recall Dynkin's definition of "regular semisimple subalgebra" of a complex semisimple Lie algebra (say $\mathfrak{g}$), which has not become standard in later literature. This just means a semisimple subalgebra of $\mathfrak{g}$ which contains a Cartan subalgebra (sometimes called a "maximal toral subalgebra" in this special case, any two being conjugate under the adjoint group). Since the subalgebra is semisimple, it involves a symmetric set of roots of $\mathfrak{g}$. The tricky step is to determine all such finite sets of roots and their conjugacy under the Weyl group. There are of course obvious examples, determined by subsets of a fixed set of simple roots: these subalgebras are now usually called "Levi subalgebras" of parabolic subalgebras. Other examples are more subtle, such as type $A_2$ inside $\mathfrak{g}$ of type $G_2$.

The paper by Dynkin which you mention appeared first in 1952 in Russian here, but an English translation has apparently only been published in an AMS translation volume (with four other papers) in 1957. By now this article is somewhat old-fashioned, of course, but along with Dynkin's other early work it has had a lot of influence on later work.

An early (independent) approach by Borel and de Siebenthal was announced in 1948, with full details provided in a longer paper in the Swiss journal Comment. Math. Helv. 23 (1949), 200-222. This paper is in French and relies on the extended Coxeter-Dynkin diagram attached to a compact semisimple Lie group. Here one has to know that such Lie groups correspond precisely to complex semisimple Lie groups or their Lie algebras. (It was shown in the late 1950s by Chevalley that such Lie groups over $\mathbb{C}$ correspond naturally to semisimple algebraic groups, which opens up the algebraic approaches further.)

Dynkin himself returned to Lie theory only sporadically, working out another approach with a collaborator here. Besides this treatment, there are by now a lot of differing accounts of the classical semisimple theory in textbooks and articles, but for example the details you ask for in 2) may be hard to locate in precisely that form. In any event, I'd urge you to get familiar with some of the later literature, which may be easier to read than the translation of Dynkin's paper.