  We introduce and study "2-roots", which are symmetrized tensor products of
orthogonal roots of Kac--Moody algebras. We concentrate on the case where $W$
is the Weyl group of a simply laced Y-shaped Dynkin diagram $Y_{a,b,c}$ having
$n$ vertices and with three branches of arbitrary finite lengths $a$, $b$ and
$c$; special cases of this include types $D_n$, $E_n$ (for arbitrary $n \geq
6$), and affine $E_6$, $E_7$ and $E_8$. We show that a natural codimension-$1$
submodule $M$ of the symmetric square of the reflection representation of $W$
has a remarkable canonical basis $\mathcal{B}$ that consists of 2-roots. We
prove that, with respect to $\mathcal{B}$, every element of $W$ is represented
by a column sign-coherent matrix in the sense of cluster algebras. If $W$ is a
finite simply laced Weyl group, each $W$-orbit of 2-roots has a highest
element, analogous to the highest root, and we calculate these elements
explicitly. We prove that if $W$ is not of affine type, the module $M$ is
completely reducible in characteristic zero and each of its nontrivial direct
summands is spanned by a $W$-orbit of 2-roots.
