The groups $Sp(4n+1)$ and $Spin(8n-2)$ as framed manifolds

We consider a compact Lie group as a framed manifold equipped with the left invarianat framing $\mathcal{L}$. In a previous paper we have proved that the Adams $e_{\mathbb{C}}$-invariant value of $SU(2n)$ $(n\ge 2)$ gives a generator of the image of $e_{\mathbb{C}}$ by twisting $\mathcal{L}$ by a certain map. In this note we show that in a similar way we can obtain analogous results for $Sp(4n+1)$ and $Spin(8n-2)$ $(n\ge 1)$.