The starting point is that the $3$-vector $\vec{\bf c}$ transforms in the $3$-dimensional irreducible vector representation of the rotation group $SO(3)$,

$$ \{ \vec{\bf c}, \vec{\bf L}\cdot \hat{\bf n} \}_{PB}~=~ \hat{\bf n}\times \vec{\bf c},$$

where $\hat{\bf n}$ is an arbitrary unit vector, whose Poisson bracket (PB) with anything vanishes

$$ \{ \hat{\bf n}, \cdot \}_{PB}~=~0.$$

We assume that $\vec{\bf c}$ is not identically zero. Since the PB with $\vec{\bf c}$ does not vanish, the $3$-vector $\vec{\bf c}$ cannot be a constant. It must be a function of the phase space variables $\vec{\bf r}$ and $\vec{\bf p}$. It can be thought of as being of the form

$$\vec{\bf c}~=~\vec{\bf r}\cdot\vec{\bf f}+ \vec{\bf p}\cdot\vec{\bf g}+ \vec{\bf L}\cdot\vec{\bf h},$$

where

$$\vec{\bf f}~=~\vec{\bf f}(r^2,p^2,\vec{\bf r}\cdot\vec{\bf p},L^2), \quad\vec{\bf g}~=~\vec{\bf g}(r^2,p^2,\vec{\bf r}\cdot\vec{\bf p},L^2), \quad\text{and}\quad \vec{\bf h}~=~\vec{\bf h}(r^2,p^2,\vec{\bf r}\cdot\vec{\bf p},L^2), $$ are three suitable vector-valued functions of the phase space $SO(3)$ scalars

$$r^2,\quad p^2,\quad \vec{\bf r}\cdot\vec{\bf p},\quad\text{and}\quad L^2.$$

References:

1. H. Goldstein, Classical Mechanics; Section 9-6 in 2nd edition or Section 9.7 in 3rd edition.

The starting point is that the $3$-vector $\vec{\bf c}$ transforms in the $3$-dimensional irreducible vector representation of the rotation group $SO(3)$,

$$\tag{1} \{ \vec{\bf c}, \vec{\bf L}\cdot \hat{\bf n} \}_{PB}~=~ \hat{\bf n}\times \vec{\bf c},$$

where $\hat{\bf n}$ is an arbitrary unit vector, whose Poisson bracket (PB) with anything vanishes

$$\tag{2} \{ \hat{\bf n}, \cdot \}_{PB}~=~0.$$

We assume that $\vec{\bf c}$ is not identically zero. Since the PB with $\vec{\bf c}$ does not vanish, the $3$-vector $\vec{\bf c}$ cannot be a constant. It must be a function of the phase space variables $\vec{\bf r}$ and $\vec{\bf p}$. It can be thought of as being of the form

$$\tag{3} \vec{\bf c}~=~\vec{\bf r}f+ \vec{\bf p}g+ \vec{\bf L}h,$$

where

$$\tag{4} f~=~f(r^2,p^2,\vec{\bf r}\cdot\vec{\bf p},L^2), \quad g~=~g(r^2,p^2,\vec{\bf r}\cdot\vec{\bf p},L^2), \quad\text{and}\quad h~=~h(r^2,p^2,\vec{\bf r}\cdot\vec{\bf p},L^2), $$ are three suitable functions of the phase space $SO(3)$ scalars

$$\tag{5} r^2,\quad p^2,\quad \vec{\bf r}\cdot\vec{\bf p},\quad\text{and}\quad L^2.$$

References:

1. H. Goldstein, Classical Mechanics; Section 9-6 in 2nd edition or Section 9.7 in 3rd edition.

The upshot is that the $3$-vector $\vec{\bf c}$ transform in the $3$-dimensional irreducible vector representation of the rotation group $SO(3)$. Since $\vec{\bf c}$ is a function of the phase space variables $\vec{\bf r}$ and $\vec{\bf p}$, it can be thought of as being of the form

$$\vec{\bf c}~=~\vec{\bf r}\cdot\vec{\bf f}+ \vec{\bf p}\cdot\vec{\bf g}+ \vec{\bf L}\cdot\vec{\bf h},$$

where  $$\vec{\bf f}~=~\vec{\bf f}(r^2,p^2,\vec{\bf r}\cdot\vec{\bf p},L^2), \quad\vec{\bf g}~=~\vec{\bf g}(r^2,p^2,\vec{\bf r}\cdot\vec{\bf p},L^2), \quad\text{and}\quad \vec{\bf h}~=~\vec{\bf h}(r^2,p^2,\vec{\bf r}\cdot\vec{\bf p},L^2), $$ are three arbitrary vector-valued functions of the phase space scalars  $$r^2,\quad p^2,\quad \vec{\bf r}\cdot\vec{\bf p},\quad\text{and}\quad L^2.$$

The unit vector $\hat{\bf n}$, on the other hand, is just a constant.

The starting point is that the $3$-vector $\vec{\bf c}$ transforms in the $3$-dimensional irreducible vector representation of the rotation group $SO(3)$,

$$ \{ \vec{\bf c}, \vec{\bf L}\cdot \hat{\bf n} \}_{PB}~=~ \hat{\bf n}\times \vec{\bf c},$$

where $\hat{\bf n}$ is an arbitrary unit vector, whose Poisson bracket (PB) with anything vanishes

$$ \{ \hat{\bf n}, \cdot \}_{PB}~=~0.$$

We assume that $\vec{\bf c}$ is not identically zero. Since the PB with $\vec{\bf c}$ does not vanish, the $3$-vector $\vec{\bf c}$ cannot be a constant. It must be a function of the phase space variables $\vec{\bf r}$ and $\vec{\bf p}$. It can be thought of as being of the form

$$\vec{\bf c}~=~\vec{\bf r}\cdot\vec{\bf f}+ \vec{\bf p}\cdot\vec{\bf g}+ \vec{\bf L}\cdot\vec{\bf h},$$

where 

$$\vec{\bf f}~=~\vec{\bf f}(r^2,p^2,\vec{\bf r}\cdot\vec{\bf p},L^2), \quad\vec{\bf g}~=~\vec{\bf g}(r^2,p^2,\vec{\bf r}\cdot\vec{\bf p},L^2), \quad\text{and}\quad \vec{\bf h}~=~\vec{\bf h}(r^2,p^2,\vec{\bf r}\cdot\vec{\bf p},L^2), $$ are three suitable vector-valued functions of the phase space $SO(3)$ scalars 

$$r^2,\quad p^2,\quad \vec{\bf r}\cdot\vec{\bf p},\quad\text{and}\quad L^2.$$

References:

1. H. Goldstein, Classical Mechanics; Section 9-6 in 2nd edition or Section 9.7 in 3rd edition.