I have read in several places that the total Pontryagin classes of real vector bundles satisfy a Whitney sum formula $p(E\oplus F) = p(E)\cdot p(F)$ modulo 2-torsion. I would like to understand this better, and have two precise questions.

1. Using the definition $p_i(E)=(-1)^{i}c_{2i}(E_\mathbb{C})$, the fact that complexification respects direct sums, and the Whitney sum formula for Chern classes, I find for example that $$ p_1(E\oplus F) = -c_2(E_\mathbb{C}\oplus F_\mathbb{C}) = -c_2(E_\mathbb{C}) - c_1(E_\mathbb{C})c_1(F_\mathbb{C}) - c_2(F_\mathbb{C}), $$ and $$p_1(E) + p_1(F)=-c_2(E_\mathbb{C})-c_2(F_\mathbb{C}).$$ I do not see why the difference $c_1(E_\mathbb{C})c_1(F_\mathbb{C})$ between these two expressions is necessarily 2-torsion. What am I missing?
2. edit: I've moved the second question to a separate post Whitney sum formula for Pontryagin classes II

I have read in several places that the total Pontryagin classes of real vector bundles satisfy a Whitney sum formula $p(E\oplus F) = p(E)\cdot p(F)$ modulo 2-torsion. I would like to understand this better, and have two precise questions.

1. Using the definition $p_i(E)=(-1)^{i}c_{2i}(E_\mathbb{C})$, the fact that complexification respects direct sums, and the Whitney sum formula for Chern classes, I find for example that $$ p_1(E\oplus F) = -c_2(E_\mathbb{C}\oplus F_\mathbb{C}) = -c_2(E_\mathbb{C}) - c_1(E_\mathbb{C})c_1(F_\mathbb{C}) - c_2(F_\mathbb{C}), $$ and $$p_1(E) + p_1(F)=-c_2(E_\mathbb{C})-c_2(F_\mathbb{C}).$$ I do not see why the difference $c_1(E_\mathbb{C})c_1(F_\mathbb{C})$ between these two expressions is necessarily 2-torsion. What am I missing?
2. edit: I've moved the second question to a separate post Whitney sum formula for Pontryagin classes II

I have read in several places that the total Pontryagin classes of real vector bundles satisfy a Whitney sum formula $p(E\oplus F) = p(E)\cdot p(F)$ modulo 2-torsion. I would like to understand this better, and have two precise questions.

1. Using the definition $p_i(E)=(-1)^{i}c_{2i}(E_\mathbb{C})$, the fact that complexification respects direct sums, and the Whitney sum formula for Chern classes, I find for example that $$ p_1(E\oplus F) = -c_2(E_\mathbb{C}\oplus F_\mathbb{C}) = -c_2(E_\mathbb{C}) - c_1(E_\mathbb{C})c_1(F_\mathbb{C}) - c_2(F_\mathbb{C}), $$ and $$p_1(E) + p_1(F)=-c_2(E_\mathbb{C})-c_2(F_\mathbb{C}).$$ I do not see why the difference $c_1(E_\mathbb{C})c_1(F_\mathbb{C})$ between these two expressions is necessarily 2-torsion. What am I missing?
2. Is there a reference which describes the difference between $p(E\oplus F)$ and $p(E)\cdot p(F)$, perhaps in terms of Bocksteins of Stiefel-Whitney classes of $E$ and $F$?

I have read in several places that the total Pontryagin classes of real vector bundles satisfy a Whitney sum formula $p(E\oplus F) = p(E)\cdot p(F)$ modulo 2-torsion. I would like to understand this better, and have two precise questions.

1. Using the definition $p_i(E)=(-1)^{i}c_{2i}(E_\mathbb{C})$, the fact that complexification respects direct sums, and the Whitney sum formula for Chern classes, I find for example that $$ p_1(E\oplus F) = -c_2(E_\mathbb{C}\oplus F_\mathbb{C}) = -c_2(E_\mathbb{C}) - c_1(E_\mathbb{C})c_1(F_\mathbb{C}) - c_2(F_\mathbb{C}), $$ and $$p_1(E) + p_1(F)=-c_2(E_\mathbb{C})-c_2(F_\mathbb{C}).$$ I do not see why the difference $c_1(E_\mathbb{C})c_1(F_\mathbb{C})$ between these two expressions is necessarily 2-torsion. What am I missing?
2. edit: I've moved the second question to a separate post Whitney sum formula for Pontryagin classes II