  Let $G$ be a complex connected simple algebraic group with a fixed real form
$\sigma$. Let $G(\mathbb{R})=G^\sigma$ be the corresponding group of real
points. This paper reports a finiteness theorem for the classification of
irreducible unitary Harish-Chandra modules of $G(\mathbb{R})$ (up to
equivalence) having non-vanishing Dirac cohomology. Moreover, we study the
distribution of the spin norm along Vogan pencils for certain $G(\mathbb{R})$,
with particular attention paid to the unitarily small convex hull introduced by
Salamanca-Riba and Vogan.
