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.



A conjecture about parallelizable generalized spheres
It is true in general that $E\_d$ is trivial. As remarked by Neil Strickland above, this boils down to showing that $TS^d\oplus\Lambda^2 TS^d$ is always a trivial bundle. To see this, represent $S^d$ as $SO(d+1)/SO(d)$. Then all bundles in question are homogeneous vector bundles (and the isomorphisms used in the comment are compatible with the homogeneous structure). These homogeneous bundles are classified by representations of $SO(d)$ and the representation in question is $\mathbb R^d\oplus\Lambda^2\mathbb R^d\cong \mathbb R^d\oplus\mathfrak{so}(d)$. But as a representation of $SO(d)$, this is isomrophic to the restriction of the adjoint representation $\mathfrak{so}(d+1)$ of $SO(d+1)$. This implies that $TS^d\oplus\Lambda^2 TS^d$ actually is an associated bundle to the the principal bundle $SO(d+1)\times\_{SO(d)}SO(d+1)\to SO(d+1)/SO(d)$ and this principal bundle is easily seen to be trivial (mapping $(g,g')\to (gSO(d),gg')$ factorises to an isomorphism from the associated bundle to $(SO(d+1)/SO(d))\times SO(d+1)$). 


This actually is an istance of a so-called tractor bundle over the homogeneous model of a Cartan geometry. There is an anlogous result associated to any representation of $SO(d+1)$. 


