Why is the transition from 3P1 to 3S1 in Helium forbidden?

[Chaosmarch]

The transition in the n=2 state of Helium from 3P1 (m=0) to 3S1 (m=0) is forbidden. I know this, because I went to a lecture where a group measured the fine structure constant very precisely using this fact. However, as far as I can see, according to atomic selection rules (available here: http://en.wikipedia.org/wiki/Selection_rule#Summary_table) this should be a perfectly acceptable E1 transition. Can anyone explain why this transition is forbidden?

For reference, the 3P1 (m=0) to 3S1 (m=+/-1) transition is allowed.

 

[Meir Achuz]

The Clebsch Gordan coefficient for 1+1=1 with all m=0 vanishes.
This is because 1+1=1 is antisymmetric.

 

[Chaosmarch]

So the selection rules in that table do not account for matrix elements vanishing? I was under the impression the selection rules were derived from the CG coefficients.

 

[Meir Achuz]

Wikipedia just left some out. Not unusual for W.