On homotopy elements represented by quotients of Lie groups

Abstract

Consider the quotient $G/B$ of a simple matrix Lie group $G$ by a subgroup $B$ isomorphic to a direct product of some of $S^1$s and $S^3$s such that its adjoint representation can be extended over $G$. Then it naturally inherits a stable framing from a twisted left invariant framing $\mathscr{L}^\alpha$ of $G$ where $\alpha$ is the realization of a complex representation of $G$. In this note we want to add some homotopy elements represented by such quotient framed manifolds to those presented in a table of [E. Ossa 1982].