While I was studying Ch 2.6 of Weinberg QFT vol I, I came to have some question about the conservation of parity and time reversal. In page 74 and 75 of Weinberg vol I, it says the fact that fundamental multiplication rule of the Poincare group is still valid even when we concerns improper Lorentz transformations(such as P and T in the beginning of page 75) incorporate that P or T are conserved. However, we know that Parity and Time Reversal are not conserved in real physics, so fundamental multiplication rule of the Poincare group is not valid when we concerns improper Lorentz transformations. But, I don't know why.. fundamental multiplication rule is from pure group theoretical approach. isn't it? Please point out what I am missing.