What is the proof that S0_O(3,1) is connected?

[latentcorpse]

define the group
$S0_O(3,1) = \{ a \in SO(3,1) | (ae_4,e_4) < 0 \}$
i need to show this group is connected

in my notes it says a group G is connected if it satisfies any of the following:
(i)any two elements of G can be joined by a $C^k$-path in G
(ii) it is not the disjoint union of two non-empty open sets
(iii) it is generated by a neighbourhood of 1 (the identity matrix)
(iv) it is generated by $exp \mathfrak{g}$

im not sure which one to try and prove or how to go about it really. can anybody offer some advice? thanks.

 

[Tinyboss]

I don't know enough about SO(3,1) to figure out the answer, but another potential way to show it connected is to show it's the image of a known-connected space under a continuous map.

 

[latentcorpse]

bump.

ive got a feeling its going to be a proof by contradiction but that might be a whole load of rubbish.