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In this note we study in a finite dimensional Lie algebra ${\mathfrak g}$ theset of all those elements x for which the closed convex hull of the adjointorbit contains no affine lines; this contains in particular elements whoseadjoint orbits generates a pointed convex cone~$C_x$. Assuming that ${\mathfrakg}$ is admissible, i.e., contains a generating invariant convex subset notcontaining 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 studypairs $(x,h)$ 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 away that ${\rm ad} h$ defines a $5$-grading, and characterize the cases wherewe even get a $3$-grading.

Elements in pointed invariant cones in Lie algebras and corresponding affine pairs