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Move Analysis

By just moving back say to e7, you have given the opponent move a3 for free, which is no big deal.
By taking 4...Bxc3+ 5 bxc3 dxe4 6 dxe4 Qxd1+ 7 Kxd1, you go into a favourable endgame as the white pawns are weak.

So you take because it gives you an advantage.
@Breaking-Limits I think what is crucial in this position is the fact that after exchanging the Bishop for Knight, black can exchange the pawn on e4. This exchange leaves white with doubled isolated pawns on the c-File, which for sure are a weakness for white.
White gets the two bishops on an open board, which I consider more than enough compensation for the broken pawn structure. The double c-pawns control lots of squares on the d-file, and the king will find safety on f2--after the f-pawn advances to f3--or even e2 after the knight is developed. The only advantage black gets is an outpost on c4. I would consider Bxc3 a mistake.
#6 4...Bxc3+ 5 bxc3 dxe4 6 Qg4 Qf6 defending g7 and attacking the weak pawn c3.

#5 I beg to disagree. Yes, the bishop's pair offers white some compensation. Yes the white king can find a home at f2 or e2, but that takes 3 moves. Meanwhile black easily develops and puts rooks on open files. White will have to oppose and trade rooks. An endgame is in sight where the doubled, isolated pawns c2, c3, a3 are weak and where black can put a knight or later his king on the weak square c4.
I believe 4...Bxc3+ is the right move. Black gets a small but distinctive advantage. What more can you wish with black after move 4?

Also if you play 3...Bb4, it is with the intent to sacrifice the minor exchange 4...Bxc3 in response to 4 a3. If black is not willing to give up his bishop for the knight, then 3...Be7 makes more sense than 3...Bb4.

Black also captures the bishop in the main line French, i.e. with 2 d4 instead of 2 d3: 1 e4 e6 2 d4 d5 3 Nc3 Bb4 4 a3 Bxc3+ 5 bxc3 dxe4 6 Qg4. In the French with 2 d3 black usually avoids the pin 3...Bb4: 1 e4 e6 2 d3 d5 3 Nd2. Even in the main line French it was usual to avoid the pin 3...Bb4 with the Tarrasch Variation: 1 e4 e6 2 d4 d5 3 Nd2.
I have to agree with tpr here. The bishop pair offers enough compensation for the weakened pawn structure sure, but that doesn't mean that Bxc3 is bad. It certainly is better than the alternative Be7 where Black gives White a move and will likely be playing with a space or development disadvantage. It's not like the bishop pair will devastate Black's position. They don't really have excellent diagonals. After Bxc3 and dxe5 Black is very solid and his bishop can go to e6 and knight to c5 or c4. Black is very comfortable, and it's probably a draw once the light squared bishops inevitably get traded, but White's weak structure means that Black can press in an endgame.
Its simple. If you dont want to take the knight after a3 dont play bishop b4. Sure there are exceptions, but 90 percent of the time a3 is just a free move for white. So if you weren't planning on capturing the knight if provoked, play bishop e7 right away.

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