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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.

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    $\begingroup$ SPLAG has information about hyperbolic root systems (signature $(n-1,1)$) but I don't remember discussion of the $(n-2,2)$ case there. $\endgroup$
    – Noam D. Elkies
    Commented Feb 5, 2017 at 2:55

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