Dickson invariant

For an orthogonal group
Suppose $$K$$ is a field, $$n$$ is a natural number, and $$b$$ is a nondegenerate symmetric bilinear form for the $$n$$-dimensional vector space $$K^n$$. Let $$G$$ be the orthogonal group corresponding to $$b$$.

The Dickson invariant is a group homomorphism from $$G$$ to cyclic group:Z2 (which we think of as a integers modulo 2) defined in the following equivalent ways:


 * 1) The Dickson invariant of $$g$$ is the parity (i.e., value mod 2) of the rank of the linear transformation $$I - g$$ where $$I$$ is the identity transformation. In other words, if $$I - g$$ has even rank, the Dickson invariant is 0 mod 2 (the trivial element of the group of integers mod 2). If $$I - g$$ has odd rank, the Dickson invariant is 1 mod 2.
 * 2) In the case that $$G$$ is generated by reflections, the Dickson invariant is the parity of the number of reflections that need to be multiplied to get $$g$$.

If the characteristic of $$K$$ is not 2, then the determinant map and Dickson invariant are equivalent in the following sense: the Dickson invariant is 0 mod 2 iff the determinant is 1 as an element of $$K$$, and the Dickson invariant is 1 mod 2 iff the determinant is -1 as an element of $$K$$.

If the characteristic of $$K$$ is 2, then the determinant of any element in any orthogonal group is always 1 as an element of $$K$$, so the Dicksoon invariant carries more information than the determinant.