  K. Saito (Publ. RIMS 21 (1), 1985, 75-179) has introduced a class of root
systems called elliptic root systems which lies in the real vector space $F$
with a metric $I$ whose signature is of type $(l,2,0)$. He also classified the
pair $(R,G)$ of an elliptic root system $R$ with one dimensional subspace $G$
of the radical of $I$, under the assumption that the quotient root system $R/G$
is reduced. Here, we classify the pairs $(R,G)$ where $R/G$ is non-reduced.
