# at.algebraic topology – \$pi_{2n-1}(operatorname{SO}(2n))\$ element represents the tangent bundle \$TS^{2n}\$, not torsion and indivisible?

Question: Is the element $$alpha$$ in $$pi_{2n-1}(operatorname{SO}(2n))$$ representing the tangent bundle $$TS^{2n}$$ of the sphere $$S^{2n}$$ indivisible and not torsion?

My understanding so far —

An $$operatorname{SO}(2n)$$ bundle over $$S^{2n}$$ corresponds to an element in $$pi_{2n}operatorname{BSO}(2n) =pi_{2n-1}operatorname{SO}(2n)$$.

Not torsion: There does not exist any integer $$m > 0$$ such that $$malpha$$ is a trivial element.

Indivisible: There does not exist any integer $$k > 1$$ and any element $$beta$$ in $$pi_{2n-1}operatorname{SO}(2n)$$ such that $$alpha=kbeta$$.

Ref: Mimura, Toda: Topology of Lie groups. Chapter IV Corollary 6.14.