  For a general cubic fourfold $X\subset\mathbb{P}^5$ with Fano scheme of lines
$F$, we prove a number of properties of the universal family of lines $I\to F$
and various subloci. We first describe the moduli and ramification theory of
the genus four fibration $p:I\to X$ and explore its relation to a birational
model of $F$ in $I$. The main part of the paper is devoted to describing the
locus $V\subset F$ of triple lines, i.e., the fixed locus of the Voisin map
$\phi:F\dashrightarrow F$, in particular proving it is an irreducible
projective singular surface of class $21\mathrm{c}_2(\mathcal{U}_F)$ and
detailing its intersection with the locus $S$ of second type lines. A
consequence of the analysis of the singularities of $V$ is a geometric proof of
the fact that if $X$ is very general, then the number of singular (necessarily
1-nodal) rational curves in $F$ of primitive class is 3780.
