How come that in various articles about pairings I never saw anybody mention that there is a possibility to turn Type 2 pairing into Type 1 by setting $e' : G_2 \times G_2 \rightarrow \mu, (P,Q) \mapsto e(Tr(P), Q)$, where $e : G_1 \times G_2 \rightarrow \mu$ is Type 2 pairing and $Tr$ is the trace map?
The only drawbacks I can see is that points from $G_2$ have rather nasty representation and there is performance loss associated with the need to calculate $Tr$. Is it enough to not consider such object or are there other problems with this approach?
