AbstractIn this note, we study in a finite dimensional Lie algebra $${\mathfrak g}$$
g
the set of all those elements x for which the closed convex hull of the adjoint orbit contains no affine lines; this contains in particular elements whose adjoint orbits generates a pointed convex cone $$C_x$$
C
x
. Assuming that $${\mathfrak g}$$
g
is admissible, i.e., contains a generating invariant convex subset not containing affine lines, we obtain a natural characterization of such elements, also for non-reductive Lie algebras. Motivated by the concept of standard (Borchers) pairs in QFT, we also study pairs (x, h) of Lie algebra elements satisfying $$[h,x]=x$$
[
h
,
x
]
=
x
for which $$C_x$$
C
x
pointed. Given x, we show that such elements h can be constructed in such a way that $$\mathop {\mathrm{ad}}\nolimits h$$
ad
h
defines a 5-grading, and characterize the cases where we even get a 3-grading.
Compactly embedded Cartan algebras and invariant cones in Lie algebras
