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An elementary      n ×  n n \times n       matrix over a ring      R R       has 1 in each diagonal position and at most one additional nonzero element. Let      R = Z [ d ] R = {\mathbf {Z}}[\sqrt d ]       where      d d       is an integer less than      −  4 - 4      . We give an algorithm for determining whether or not a      2 ×  2 2 \times 2       invertible matrix over      R R       is generated by elementary matrices. This is connected with the theory of integral binary quadratic forms.

Detecting products of elementary matrices in ${\rm GL}\sb{2}({\bf Z}[\sqrt{d}])$