= Solution
A smooth <principal bundle> with structure <Lie group> $G$ can be specified by a locally trivial map $\pi:E\to B$ whose fiber coordinate changes have the form $(x,g)_i\mapsto(x,h_{ji}(x)g)_j$, with the usual <transition functions of a principal bundle> cocycle on triple overlaps. In each chart define $(x,g)\cdot k=(x,gk)$. On overlaps, $h_{ji}(x)(gk)=(h_{ji}(x)g)k$, so these local right actions agree. They therefore give a global smooth right <group action>, free and transitive on every fiber. Equivalently, this action together with equivariant local triviality is the definition of a <principal bundle>.
Use five-dimensional <Minkowski spacetime> with ambient metric $\eta=\operatorname{diag}(-1,1,1,1,1)$ and realize unit-radius <de Sitter spacetime> as $\eta(x,x)=1$. Its <tangent space> is $x^\perp$, of Lorentzian signature $(3,1)$. An oriented <pseudo-orthonormal frame> $(e_0,e_1,e_2,e_3)$ of $x^\perp$, together with the last column $x$, is precisely a matrix
$$
g=(e_0,e_1,e_2,e_3,x)\in SO(4,1).
$$
Conversely the last column of such a matrix lies on the hyperboloid, and its other columns give that frame. Thus
$$
\boxed{SO(4,1)\longrightarrow\mathrm{dS}_4,
\qquad g\longmapsto ge_4,\qquad G=SO(3,1).}
$$
Right multiplication by $\operatorname{diag}(h,1)$ fixes $x$ and changes the tangent frame by $h$. It is free and transitive on frames over $x$, proving the <principal bundle> description. Smooth local choices of frames provide its local trivializations. The dimensions are $10=4+6$.
This is the <oriented frame bundle of de Sitter spacetime>. Allowing orientation reversal gives $O(4,1)$ and structure group $O(3,1)$; imposing a time orientation as well selects $SO_0(4,1)$ and $SO_0(3,1)$. For radius $a$, take the last column $x/a$.
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