The Radon-Hurwitz number $k(n)$ is the largest $k$ such that there exists an orthogonal multiplication $\mathbb R^k\times \mathbb R^n\to \mathbb R^n$; so for an ONB $x_1,\dots, x_k$ of $\mathbb R^k$ and a unit vector $y\in \mathbb R^n$ the vectors $y, x_1.y, x_2.y,\dots x_k.y$ are orthogonal in $\mathbb R^n$. This describes vector fields on $S^{n-1}$. The orthogonal multiplications were constructed by Radon [Lineare Scharen orthogonaler Matrizen, Abh. Math. Sem. Univ. Hamburg 1, 1-14, 1921] who extended the construction of Hurwitz for $k=n=1,2,4,8$. They extend to representations of Clifford algebras $C(\mathbb R^k, -\langle\quad,\quad\rangle)$ which explains the "periodicity" in $k(n)$ with respect to 8 and 2. Adams showed that there are not more linearly independent vector fields on the sphere $S^{n-1}$.

The Radon-Hurwitz number $k(n)$ is the largest $k$ such that there exists an orthogonal multiplication $\mathbb R^k\times \mathbb R^n\to \mathbb R^n$; so for an ONB $x_1,\dots, x_k$ of $\mathbb R^k$ and a unit vector $y\in \mathbb R^n$ the vectors $y, x_1.y, x_2.y,\dots x_k.y$ are orthogonal in $\mathbb R^n$. This describes vector fields on $S^{n-1}$. The orthogonal multiplications were constructed by Radon [Lineare Scharen orthogonaler Matrizen, Abh. Math. Sem. Univ. Hamburg 1, 1-14, 1921] who extended the construction of Hurwitz for $k=n=1,2,4,8$. They extend to representations of Clifford algebras $C(\mathbb R^k, -\langle\quad,\quad\rangle)$ which explains the "periodicity" in $k(n)$ with respect to 8 and 2.

Radon writes: "For real matrices the solution offered itself to me by a particular reduction method using matrices whose elements are complex numbers or quaternions." (My translation) Maybe, there is an inkling of the fiber method you are looking at.

Adams showed that there are not more linearly independent vector fields on the sphere $S^{n-1}$.

The Radon-Hurwitz number $k(n)$ is the largest $k$ such that there exists an orthogonal multiplication $\mathbb R^k\times \mathbb R^n\to \mathbb R^n$; so for an ONB $x_1,\dots, x_k$ of $\mathbb R^k$ and a unit vector $y\in \mathbb R^n$ the vectors $y, x_1.y, x_2.y,\dots x_k.y$ are orthogonal in $\mathbb R^n$. This describes vector fields on $S^{n-1}$. The orthogonal multiplications were constructed by Radon [Lineare Scharen orthogonaler Matrizen, Abh. Math. Sem. Univ. Hamburg 1, 1-14, 1921] who extended the construction of Hurwitz for $k=n=1,2,4,8$. They extend to representations of Clifford algebras $C(\mathbb R^k, -\langle\quad,\quad\rangle)$. Adams showed that there are not more linearly independent vector fields on the sphere $S^{n-1}$.

The Radon-Hurwitz number $k(n)$ is the largest $k$ such that there exists an orthogonal multiplication $\mathbb R^k\times \mathbb R^n\to \mathbb R^n$; so for an ONB $x_1,\dots, x_k$ of $\mathbb R^k$ and a unit vector $y\in \mathbb R^n$ the vectors $y, x_1.y, x_2.y,\dots x_k.y$ are orthogonal in $\mathbb R^n$. This describes vector fields on $S^{n-1}$. The orthogonal multiplications were constructed by Radon [Lineare Scharen orthogonaler Matrizen, Abh. Math. Sem. Univ. Hamburg 1, 1-14, 1921] who extended the construction of Hurwitz for $k=n=1,2,4,8$. They extend to representations of Clifford algebras $C(\mathbb R^k, -\langle\quad,\quad\rangle)$ which explains the "periodicity" in $k(n)$ with respect to 8 and 2. Adams showed that there are not more linearly independent vector fields on the sphere $S^{n-1}$.