My understanding of isotropic is the a particular physics law remain same no matter at what direction I look at it? Now suppose in case of inertial frame, we know that its is homogeneous and isotropic in space and isotropic in time. So it essentially imply that any physical quantity (say $L$) is same if it's at $r$ or $r'$ for a given $x$-$y$ inertial frame $S$ from homogeneity of space. Now it also implies that by isotropic of space, given any $r$ every direction gives same $L$ (and hence $L$ is a function of $\lvert\vec{v}\rvert$). I don't get how can we comment on isotropic of $L$ for a given $S$ because once we know $S$, we essentially know the origin $O$ and hence for a given $r$, the direction to look at it is fixed i.e radius vector $\vec{r}$.
P.S My question is closely inspired from my inability to understand the propertied of $L$ given in L.D Landau Mechanics on page 5.