  In this note we study in a finite dimensional Lie algebra ${\mathfrak g}$ 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~$C_x$. Assuming that ${\mathfrak
g}$ 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 $[h,x]=x$ for which $C_x$
pointed. Given $x$, we show that such elements $h$ can be constructed in such a
way that ${\rm ad} h$ defines a $5$-grading, and characterize the cases where
we even get a $3$-grading.
