= Admissible crystallographic Coxeter graph
{title2=$G_{ii}=1,\quad G_{ij}=-\sqrt{n_{ij}}/2$}
= Admissible Coxeter graph
{synonym}
In the crystallographic convention, a <Coxeter graph> is admissible when it is realized by linearly independent unit <vectors> $v_i$ in a <Euclidean space>, with nonpositive mutual <inner products> and $4(v_i,v_j)^2=n_{ij}\in\{0,1,2,3\}$ for distinct vertices. The number $n_{ij}$ records a single, double or triple connection; equivalently, the displayed <Gram matrix> is <positive-definite>. In edge-label notation the nonzero connections have labels $3,4,6$. Forgetting multiplicity, the underlying <graph> is a <forest>: for any cycle of $k\geq3$ distinct vertices, the sum of the corresponding unit <vectors> would have squared <norm> at most $k-k=0$, contradicting their linear independence. Multiple lines on one connection are labels and are not length-two cycles. This is the convention of https://www.mathematik.uni-muenchen.de/~wehler/LieAlgebrasScript.pdf[Wehler's Lie algebra notes, section 6.3].
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