Mathematically the Lorentz group is precisely the $O(1,3)$ is the 4-vector rotation preserving the inner product of 4-vector under this metric $$ \eta_{\mu \nu}=(+1,-1,-1,-1). $$ There are four distinct sectors of this $O(1,3)$.

Say the 4-vector is $A$ and $B$, then for the Lorentz transformation $R_{n \nu} \in O(1,3)$ on the 4-vector, we have $$ A^\mu \eta_{\mu \nu} B^{\nu} ={A^\mu}' \eta_{\mu \nu} {B^\nu}'=A^\mu R^{T}_{\mu m} \eta_{m n} R_{n \nu} B^{\nu} $$

Lorentz group $O(1,3)$ explains the symmetry group of the spacetime at any fixed point.

Built in on this data of Lorentz group $O(1,3)$, how do we explain what is light cone to mathematicians?

Namely what is light cone? mathematically? How to explain to mathematicians who understand $O(1,3)$, but not light cone?

What exactly is this cone of light cone defined mathematically?