A.3.1 Elementary matrices (Chevalley generators) for Oð Þ 2l þ 1; k

Following the theory of Lie algebra, we index rows by 0, 1, …, l, � 1, …, � l. These elementary matrices are listed in Table A5.

Elementary matrices for the odd-orthogonal group in even characteristics differ from that of odd characteristics. In above table we made that distinction and listed them separately in different rows according to the characteristics of k. If char(k) is even, we can construct the elements wi, which interchanges the ith row with �i th row as follows:

$$w\_i = (I + \mathbf{e\_{0,i}} + \mathbf{e\_{-i,i}})(I + \mathbf{e\_{0,-i}} + \mathbf{e\_{i,-i}})(I + \mathbf{e\_{0,i}} + \mathbf{e\_{-i,i}}) = I + \mathbf{e\_{i,i}} + \mathbf{e\_{-i,-i}} + \mathbf{e\_{i,-i}} + \mathbf{e\_{-i,i}}.$$

Otherwise, we can construct wi, which interchanges the ith row with �i th row with a sign change in i th, � <sup>i</sup> th and 0th row in odd-orthogonal group as follows:

