Let $S^{d}$ denote the standard $d$-dimensional sphere. I heard from a physicist that from physical arguments they have been able to show that the vector bundle:

$E_{d} = TS^{d}\oplus \Lambda ^{d-2}T^{\ast}S^{d}$

is topologically trivial, meaning that it is parallelizable. The convention is that $\Lambda ^{0}T^{\ast}S^{2} = \mathbb{R}$ and $E_{1} = TS^{1}$. They call this bundles $E_{d}$ "generalized spheres".

Is this claim true? For $d=2$ it can be easily checked that it is correct.

Edit: What about the opposite question? namely which orientable, $d$-dimensional, Riemannian manifolds $M_{d}$ have the corresponding $E_{d} = TM_{d}\oplus \Lambda^{2} T^{\ast}M_{d}$ trivial? For example:

$S^{6}$ is the prototypical example of irreducible nearly Kahler manifold. Is it true that $E_{6}$ is trivial when taking $M_{6}$ to be any irreducible nearly-Kahler manifold?

$S^{7}$ is an example of nearly parallel $G_{2}$-manifold. Is it true that $E_{7}$ is trivial when taking $M_{7}$ to be any nearly-parallel $G_{2}$ manifold?

Thanks.