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3 deleted 1 characters in body

The group $Pin_-(2)$ is an example of what you're looking for.
It can be described explicitly as a subgroup of the group of unit quaternions:
$Pin_-(2)=$ { $a+bi| a^2+b^2=1$ } $\cup$ { $cj+dk| c^2+d^2=1$ } $\subset \mathbb H^\times$.

It's

Its main interesting properties are:
- The conjugation action of $\pi_0$ on its Lie algebra is non-trivial.
- All the elements of the non-identity component have a non-trivial square.

There is no Lie group that is diffeomorphic to $\mathbb Z/2\times \mathbb R$ and that shares those properties.

2 added 1 characters in body

The group $Pin_-(2)$ is an example of what you're looking for.
It can be described explicitly as a subgroup of the group of uni unit quaternions:
$Pin_-(2)=$ { $a+bi| a^2+b^2=1$ } $\cup$ { $cj+dk| c^2+d^2=1$ } $\subset \mathbb H^\times$.

It's main interesting properties are:
- The conjugation action of $\pi_0$ on its Lie algebra is non-trivial.
- All the elements of the non-identity component have a non-trivial square.

There is no Lie group that is diffeomorphic to $\mathbb Z/2\times \mathbb R$ and that shares those properties.

1

The group $Pin_-(2)$ is an example of what you're looking for.
It can be described explicitly as a subgroup of the group of uni quaternions:
$Pin_-(2)=$ { $a+bi| a^2+b^2=1$ } $\cup$ { $cj+dk| c^2+d^2=1$ } $\subset \mathbb H^\times$.

It's main interesting properties are:
- The conjugation action of $\pi_0$ on its Lie algebra is non-trivial.
- All the elements of the non-identity component have a non-trivial square.

There is no Lie group that is diffeomorphic to $\mathbb Z/2\times \mathbb R$ and that shares those properties.