  This note presents some properties of the variety of planes $F_2(X)\subset
G(3,7)$ of a cubic $5$-fold $X\subset \mathbb P^6$. A cotangent bundle exact
sequence is first derived from the remark made by Iliev and Manivel that
$F_2(X)$ sits as a Lagrangian subvariety of the variety of lines of a cubic
$4$-fold, which is a hyperplane section of $X$. Using the sequence, the Gauss
map of $F_2(X)$ is then proven to be an embedding. The last section is devoted
to the relation between the variety of osculating planes of a cubic $4$-fold
and the variety of planes of the associated cyclic cubic $5$-fold.
