Chevalley group of type D

Definition
Let $$K$$ be a field and $$n$$ be a natural number. The Chevalley group of type D denoted $$D_n(K)$$, is defined as follows:


 * 1) Start with the defining ingredient::split orthogonal group of degree $$2n$$ over $$K$$.
 * 2) Consider the intersection of the kernel of the defining ingredient::spinor norm map with the kernel of the definingi ingredient::Dickson invariant map (note that the kernel of the Dickson invariant map is the same as the defining ingredient::split special orthogonal group, i.e., the kernel of the determinant, when the characteristic is not 2)
 * 3) Consider the inner automorphism group of this intersection, i.e., quotient it out by its center.

The final answer obtained in Step (3) is denoted $$D_n(K)$$. It is also denoted $$P\Omega_{2n}^+(K)$$.

The notation $$D_n(q)$$ is used as a shorthand for $$D_n(\mathbb{F}_q)$$ where $$\mathbb{F}_q$$ is the (unique up to isomorphism) field of size $$q$$.