Why isn't 6.Bc4 considered by theory in the 5.Bxf7+ Traxler Counterattack? In the Traxler Counterattack of the Two Knights Defense, after
[fen &quot;&quot;]

1.e4 e5 2.Nf3 Nc6 3.Bc4 Nf6 4.Ng5 Bc5 5.Bxf7+ Ke7 6. Bb3 (6. Bc4)

why isn't 6.Bc4 considered as a &quot;serious&quot; move for White by theory? Only 6.Bb3 and 6.Bd5 are seen in this fairly rare line.
Comparing 6.Bb3 and 6.Bc4, the drawbacks of the latter are obvious since the bishop is not protected and can become a target to attacks:

either by a black piece: Ne5, Na5, Qc5, Qd4...

or, more probably, by the d-pawn: the typical ...d5 will gain a tempo for development


On the other hand, I can see no advantage of having the bishop on c4 rather than b3. You will hardly ever want to retreat to e2, nor to play a queenside fianchetto.
Given how important each tempo can be in such an complicated line, it is understandable than no strong player has ever ventured 6.Bc4. Probably there isn't a direct refutation, but you might regret having your bishop exposed later on.