  This article explores special Lagrangian submanifolds in $\mathbb{CP}^3$,
viewed as a nearly K\"ahler manifold, from two different perspectives.
Intrinsically, using a moving frame set-up, and extrinsically, using
$\mathrm{SU}(2)$ moment-type maps. We describe new homogeneous examples, from
both perspectives, and classify totally geodesic special Lagrangian
submanifolds. We show that every special Lagrangian in $\mathbb{CP}^3$, or the
flag manifold $\mathbb{F}_{1,2}(\mathbb{C}^3)$ admitting a symmetry of an
$\mathrm{SU}(2)$ subgroup of nearly K\"ahler automorphisms is automatically
homogeneous.
