The generalized orthogonal group $\text{O}(n,k)$ is the group of $(n+k)\times (n+k)$ real-valued matrices that preserve the symmetric bilinear form \([x,y]_{n,k} = x_1y_1 + \cdots + x_ny_n - x_{n+1}y_{n+1} - \cdots - x_{n+k}y_{n+k}.\)That is, $A \in \text{O}(n,k)$ if and only if $[x,y]{n,k} = [Ax,Ay]{n,k}$ for all $x,y \in \mathbb R^n$.
This group is a matrix Lie group because it is a subgroup of the general linear group $\text{GL}(n,\mathbb C)$.