  For a positive integer $s$, a lattice $L$ is said to be $s$-integrable if
$\sqrt{s}\cdot L$ is isometric to a sublattice of $\mathbb{Z}^n$ for some
integer $n$. Conway and Sloane found two minimal non $2$-integrable lattices of
rank $12$ and determinant $7$ in 1989. We find two more ones of rank $12$ and
determinant $15$. Then we introduce a method of embedding a given lattice into
a unimodular lattice, which plays a key role in proving minimality of non
$2$-integrable lattices and finding candidates for non $2$-integrable lattices.
