Dz. Dokovic et Nq. Thang, CONJUGACY CLASSES OF MAXIMAL TORI IN SIMPLE REAL ALGEBRAIC-GROUPS ANDAPPLICATIONS, Canadian journal of mathematics, 46(4), 1994, pp. 699-717
Let G be an almost simple complex algebraic group defined over R, and
let G(R) be the group of real points of G. We enumerate the G(R)-conju
gacy classes of maximal R-tori of G. Each of these conjugacy classes i
s also a single G(R)-degree-conjugacy class, where G(R)-degree is the
identity component of G(R), viewed as a real Lie group. As a consequen
ce we also obtain a new and short proof of the Kostant-Sugiura's theor
em on conjugacy classes of Cartan subalgebras in simple real Lie algeb
ras. A connected real Lie group P is said to be weakly exponential (w.
e.) if the image of its exponential map is dense in P. This concept wa
s introduced in [HM] where also the question of identifying all w.e. a
lmost simple real Lie groups was raised. By using a theorem of A. Bore
l and our classification of maximal R-tori we answer the above questio
n when P is of the form G(R)-degree.