Appendix A. Solving the word problem in G

In computational group theory, one is always looking for algorithms that solve the word problem. When G is a special linear group, one has a well-known algorithm to solve the word problem—the Gaussian elimination algorithm. One observes that the effect of multiplying an element of the special linear group by an elementary matrix (also known as elementary transvection) from left or right is either a row or a column operation, respectively. Using this algorithm one can start with any matrix g ∈SLð Þ l þ 1; k and get to the identity matrix, thus writing g as a product of elementary matrices ([18], Proposition 6.2). One of the objective of this appendix is to discuss a similar algorithm for orthogonal and symplectic groups, with a set of generators that we will call elementary matrices in their respective groups. Similar algorithms can be found in the works of Brooksbank [19, 20] and Costi [21]. However, we have no restrictions on the cardinality or characteristic of the field k.

We first describe the elementary matrices and the row-column operations for the respective groups. These row-column operations are nothing but multiplication by elementary matrices from left and right, respectively. Here elementary matrices used are nothing but Chevalley generators which follows from the theory of Chevalley groups.

The basic idea of the algorithm is to use the fact that multiplying any orthogonal matrix by any one of the generators enables us to perform row or column operations. The relation Tgβ<sup>g</sup> <sup>¼</sup> <sup>β</sup> gives us some compact relations among the blocks of <sup>g</sup> which can be used to make the algorithm faster. To make the algorithm simple, we will write the algorithm for O 2ð Þ l þ 1; k , Oþð Þ 2l; k , and O�ð Þ 2l; k separately.

#### A.1 Groups in which Gaussian elimination works

	- Since non-degenerate symmetric bilinear forms over a finite field of odd characteristics are classified ([22], p. 79) according to the β (see Section 3), we have a Gaussian elimination algorithm for all orthogonal groups over a finite field of odd characteristics.
	- Since non-degenerate quadratic forms over a perfect field of even characteristics can be classified ([23], p. 10) according to quadratic forms Q(x) defined in ([24], Section 4.2), we have a Gaussian elimination

algorithm for all orthogonal groups over a perfect field of even characteristics.

• Furthermore, we have Gaussian elimination algorithm for orthogonal groups that are given by the above bilinear forms or quadratic forms over arbitrary fields. This algorithm also works for bilinear or quadratic forms that are equivalent to the above forms.
