I apologize for asking questions that seem likely to be answered in Conway & Sloane's "Sphere Packings, Lattices, and Groups" if I knew where to look.

Let $L$ be the unique* even unimodular lattice of signature $(10,2)$, and let $\Delta \subset L$ be the vectors with norm-square $2$, called the roots.

Is this infinite set the root system of a Coxeter group? If so, which one?

Also,

How can one parametrize the set of roots?

*I suppose we can take $L$ to be $E_8 \oplus \left[{0\atop 1}{1\atop 0}\right]^{\oplus 2}$, but if there's an easier description then I'm interested in that too.

If I've missed a good tag please feel free to retag as appropriate.