  We explore whether a root lattice may be similar to the lattice $\mathscr O$
of integers of a number field $K$ endowed with the inner product $(x, y):={\rm
Trace}_{K/\mathbb Q}(x\cdot\theta(y))$, where $\theta$ is an involution of $K$.
We classify all pairs $K$, $\theta$ such that $\mathscr O$ is similar to either
an even root lattice or the root lattice $\mathbb Z^{[K:\mathbb Q]}$. We also
classify all pairs $K$, $\theta$ such that $\mathscr O$ is a root lattice. In
addition to this, we show that $\mathscr O$ is never similar to a
positive-definite even unimodular lattice of rank $\leqslant 48$, in
particular, $\mathscr O$ is not similar to the Leech lattice. In appendix, we
give a general cyclicity criterion for the primary components of the
discriminant group of $\mathscr O$.
