Lie algebra of the Lorentz group
= Lie algebra of the Lorentz group
{c}
{title2=$\mathfrak o(1,3)$}
For $M=\operatorname{diag}(-1,1,1,1)$,
$$
\mathfrak o(1,3)=\{Y\in M_4(\mathbb R):Y^TM+MY=0\}.
$$
Every such matrix has the block form
$$
Y=\begin{pmatrix}0&v^T\\v&\Omega\end{pmatrix},
\qquad
v\in\mathbb R^3,\quad\Omega^T=-\Omega,
$$
so the Lie algebra and the <Lorentz group> have dimension six. At $A\in O(1,3)$, the tangent space is $A\mathfrak o(1,3)$.