Orthogonal and special orthogonal groups

Definition

The notation $SO(n,m)$ refers to the special orthogonal group of matrices that leave invariant a non-degenerate quadratic form of signature $(n,m)$ on a real vector space. Here's a more detailed breakdown:

1. Orthogonal Group: The orthogonal group, denoted as $O(n,m)$, consists of all $(n+m)\times (n+m)$ real matrices $A$ such that

where $A^T$ is the transpose of $A$ and $g$ is a diagonal matrix with $n$ entries of +1 and $m$ entries of -1. These matrices preserve the quadratic form given by $g$.

2. Special Orthogonal Group: The special orthogonal group $SO(n,m)$ is the subgroup of $O(n,m)$ that consists of matrices with determinant +1. In other words, these are the "volume-preserving" orthogonal transformations.

For $SO(n)$ (or $SO(n,0)$), which is a common special case, the quadratic form is the standard dot product in ${\mathbb{R}}^n$, and the group consists of $n\times n$ orthogonal matrixs with determinant +1.

The case $O(1,3)$ is Lorentz group.

Lie algebra

According to Exercise 40 in @baez1994gauge page 188:
Let $g$ be a metric of signature $(p,q)$ on ${\mathbb{R}}^n$, where $p+q=n$. Show that the Lie algebra $\mathfrak{so}(p,q)$ of $SO(p,q)$ consists of all $n\times n$ real matrices $T$ with $g(Tv,w)=-g(v,Tw)$ for all $v,w\in {\mathbb{R}}^n$. Show that the dimension of $\mathfrak{so}(p,q)$, and hence that of $SO(p,q)$, is $\frac{n(n-1)}{2}$.
Determine an explicit basis of the Lorentz Lie algebra, $\mathfrak{so}(1,3)$.

It is supposed to be done with the techniques of orthogonal matrix#The Lie algebra.