A note on orthogonal similitude groups

Let V be a vector space over the field F such that charðF Þ 61⁄4 2, and let V have a symmetric nondegenerate bilinear form. Let GOðV Þ be the orthogonal similitude group for this symmetric form, with similitude character . We prove that if g 2 GOðV Þ with ðgÞ 1⁄4 , then g 1⁄4 t1t2 where t1 is an orthogonal involution, and t2 is such that t 2 2 1⁄4 I and ðt2Þ 1⁄4 . As an application, we obtain an expression for the sum of the degrees of the irreducible characters of GOðn,FqÞ for odd q.