= Component quotient of the real Lorentz group in two dimensions
{title2=$O(1,1)/SO^+(1,1)\cong C_2\times C_2$}
Let $B(x)=\begin{pmatrix}\cosh x&-\sinh x\\-\sinh x&\cosh x\end{pmatrix}$, $T=\operatorname{diag}(1,-1)$ and $P=\operatorname{diag}(-1,1)$. Every real matrix preserving $\operatorname{diag}(1,-1)$ has a unique form $B(x)S$ with $S\in\{I,T,P,TP\}$. The boost subgroup is normal because $TB(x)T=PB(x)P=B(-x)$. Its four <cosets> form the <Klein four-group>, giving the displayed <quotient group>. In contrast, the four-element subgroup generated by $T,P$ is not normal: $B(x)TB(-x)=B(2x)T$ is not diagonal for $x\ne0$. This distinguishes the normal identity component from a chosen set of component representatives.
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