$\DeclareMathOperator\SO{SO}$For $n\geq 2$, the $n$-dimensional connected Lorentz group is defined as $$\SO(n-1,1)^{\uparrow}=\{f\in M_n (\mathbb{R}):f^T\eta f=\eta, \det (f)=1,f^0_0>0\}$$ where $$\eta=\begin{bmatrix}-1 & 0 & 0& \cdots & 0\\ 0 & 1 & 0 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots &\\ 0 & 0 & 0 & 0 & 1 \end{bmatrix}.$$

Assume that $p=(E,E,0,\dotsc,0)$. I would appreciate if someone could help me about proving this fact: The stabilizer of $p$ under $\SO(n-1,1)^{\uparrow}$ is $$\SO(n-2)\ltimes \mathbb{R}^{n-2},$$ the Euclidean group. By the stabilizer of $p$ under $\SO(n-1,1)^{\uparrow}$, I mean, $\SO(n-1,1)^{\uparrow}_p =\{ f\in \SO(n-1,1)^{\uparrow}:fp=p\}$.