Letting $n=k+l$, we define the matrix $Q\in\mathrm{GL}_n(\mathbb{R})$ such that $$Q_{ij}=\begin{cases} 0 & i\neq j \\ 1 & 1\leq i=j\leq k \\ -1 & k+1\leq i=j\leq n\end{cases}$$ and we set $$\mathrm{SO}(k,l):=\{A\in\mathrm{SL}_n(\mathbb{R})\ |\ A^\mathrm{T}QA=Q\}$$ with the subspace topology given by $M_n(\mathbb{R})$. This is exercise 7.2* of Representation Theory, a First Course, by Fulton and Harris.
My usual tool for finding components is by using the determinant function, but clearly that doesn't apply here. The book suggests using the projection mappings $$\pi_1:\mathbb{R}^n\to\mathbb{R}^k,\ (v_1,\ldots,v_n)\mapsto(v_1,\ldots,v_k)$$ and $$\pi_2:\mathbb{R}^n\to\mathbb{R}^l,\ (v_1,\ldots,v_n)\mapsto(v_{k+1},\ldots,v_n)$$ and associating to any automorphism $A\in\mathrm{SO}(k,l)$ and automorphism of $\mathbb{R}^k$ and of $\mathbb{R}^l$ by means of these projections, and then looking at whether or not these resulting automorphisms preserve the orientation of $\mathbb{R}^k$ and $\mathbb{R}^l$. Firstly, though, I don't see how we even have that for $A\in\mathrm{SO}(k,l),\ A:\mathbb{R}^n\to\mathbb{R}^n$, $A\circ\pi_1$ even maps into $\mathrm{im}(\pi_1)$, and $A\circ\pi_2$ into $\mathrm{im}(\pi_2)$.
I can see that $\pi_1(v)=v$ if and only if $Qv=v$, and that $\pi_2(v)=v$ if and only if $Qv=-v$, and I suspect that this fact is necessary, but I haven't been able to successfully use this fact.
What is the correct way to solve this problem?