I am having trouble understanding a couple of lines of computation from Theorem 13.30 in Besse's Einstein Manifolds text (see image here).

We are twisting the spinor bundle (on Einstein 4-manifold) $\Sigma$ with an auxiliary bundle $S^3\Sigma^-$. We form the Dirac operator $\mathscr{D}^D$ formed by twisting the Levi-Civita connection on $\Sigma$ with a copy $D$ acting on $S^3\Sigma^-$ (and composing with the Clifford action acting trivially on $S^3\Sigma$). Besse evaluates the index of this operator using the APS theorem as an integral over Chern and Pontrjagin classes.

I understand that the $(1-\frac{1}{24}p_1)$ terms comes from the $\widehat{A}$-genus of the manifold, and further the signature theorem, $\tau=\frac13 \int_M p_1(M) $. However, I do not understand the evaluation of the Chern character of the twisting bundle as $(4-10c_2)$ and the subsequent evaluation in terms of Euler characteristic and signature.

Any insight would be greatly appreciated.

Edit: I am still not sure about the final calculation relating the index to the Euler character and signature, however, working backwards it seems we require $c_2(\Sigma^-)=\frac12e(M)-\frac14p_1(M)$ (or perhaps something a bit different if there are lower degree terms which could multiply with with the $\frac{1}{24}p_1$ term from the $\widehat{A}-$genus), where $e(M)$ is the Euler class of $M$.