Is $\operatorname{Spin}(8)$ a direct product of $\operatorname{Spin}(7)$ and $S^7$? 
Is $\textrm{Spin}(8)$ a direct product of $\textrm{Spin}(7)$ and $S^7$?

I met this statement in the literature, but without a reference. If it is true, where is it explicitly written?
 A: As I suspected, the statement that the bundle $\mathrm{SO}(8)\to S^7$ is a product bundle, i.e., that
$$
\mathrm{SO}(8)\simeq S^7\times\mathrm{SO}(7)\tag1
$$
as bundles over $S^7$ is in N. Steenrod's The topology of fibre bundles (as part of Theorem 8.6).
Note that Steenrod uses the notation $R_n$ for what is more commonly notated $\mathrm{SO}(n)$, these days.  (The '$R$' is for 'rotation group'.  He uses $O_n$ for the orthogonal group.)  His proof, based on the octonions (which he calls 'the Cayley numbers'), is exactly what I outlined in my comment above.  He does not give a reference to an earlier statement of the result, but refers to L. E. Dickson's famous Linear Algebra for the properties of the Cayley numbers that are used in the proof.
He does not discuss the spin groups, but, obviously, the equivalence of bundles
$$
\mathrm{Spin}(8)\simeq S^7\times\mathrm{Spin}(7)\tag2
$$
follows from (1) by passing to the respective double covers.
A: Robert Bryant answered this question by interpreting an element of SO(8) as an octonion multiplication. But I've understood now (after exchanging messages with Robert) that there is a more direct construction.
Let $\Gamma_{j=1,\ldots,7}$ be antisymmetric purely imaginary $8 \times 8$ matrices that satisfy the Clifford algebra
$$ \Gamma_j \Gamma_k + \Gamma_k \Gamma_j = 2\delta_{jk} \,.$$
Then 28 mutually orthogonal Hermitian generators of SO(8) are $\Gamma_j$ and $\Sigma_{jk} = i \Gamma_j \Gamma_k$.
$\Sigma_{jk}$ are also the generators of Spin(7), and it is a bit more convenient to consider instead of $SO(8)$ the so-called Semispin(8) group (see e.g.  hep-th/9906059), which is isomorphic to $SO(8)$, but rotates spinors rather than vectors. Then any element of Semispin(8) can be represented as
$$
g_8 \ =\ \exp(i\alpha_j \Gamma_j) \exp(\beta_{kl} \Gamma_k \Gamma_l) \ =\ g_{S^7} g_7,
 $$
where $g_7$ is an element of Spin(7) and $g_{S^7}$ is an embedding of $S^7$ into Semispin(8)  if restricting $\|\alpha_j\| \leq \pi$.
This already looks as a direct product searched for and the only nuissance is that the representation above is not unique: $\mathbb{1}$ of Semispin(8) may be represented either as $\mathbb{1} \times \mathbb{1}$ or as $(-\mathbb{1})\times (-\mathbb{1})$. This nuissance disappears if one goes from Semispin(8) to its double cover Spin(8). The latter is represented by two distinct matrices
$\exp(i\alpha_j \Gamma_j) $ and $g_7$, and this representation is unique.
