Split orthogonal group

Definition
This group is the defining ingredient::orthogonal group for a symmetric bilinear form where the symmetric bilinear form gives a hyperbolic space.

Let $$m$$ be a natural number and $$k$$ be any field. The split orthogonal group of degree $$2m$$ over $$k$$ can be defined as the group, under matrix multiplication:

$$\{ A \in GL(2m,k) \mid A \begin{pmatrix} 0 & I \\ I & 0 \\\end{pmatrix}A^T = \begin{pmatrix} 0 & I \\ I & 0 \\\end{pmatrix} \}$$.

Here, the $$0$$ and $$I$$ are $$n \times n$$ block matrices.

When the characteristic of $$k$$ is not equal to two, this is isomorphic to the group (in fact, they are conjugate in $$GL(2m,k)$$):

$$\{ A \in GL(2m,k) \mid A \begin{pmatrix} I & 0 \\ 0 & -I \\\end{pmatrix}A^T = \begin{pmatrix} I & 0 \\ 0 & -I \\\end{pmatrix} \}$$.

For a finite field, the split orthogonal group is also sometimes known as the orthogonal group of $$+$$ type or the $$+1$$ type.