Consider a $j=1,\,SU(2)$ representation (or fundamental $SO(3)$ representation). Suppose that $a_1, b_i, c_i$ with $i=1,2,3$ are vectors transforming under this representation i.e $a_i'=[\rho_1(\alpha)]_{ij}a_j=\rho_{ij}a_j$ and similarly for $b$ and $c$. Consider $$I_1(\vec{a},\vec{b} )=\delta_{im}a_i b_m\,\,\,\,\,\,;\,\,\,\,\,\,\,\,I_2(\vec{a}, \vec{b}, \vec{c} )=\epsilon_{imr}a_i b_m c_r$$
$1)$ Deduce the conditions on $\delta_{im}$ and $\epsilon_{imr}$ for which $I_1$ and $I_2$ are invariants under the transformation.
Attempt: $I_1(\vec{a}', \vec{b}') = a_j b_k (\rho^T \rho)_{jk} = \delta_{jk} a_j b_k$ provided $(\rho^T \rho)_{jk} = \delta_{jk}$. So is that the condition on $\delta_{ij}?$
Similarly, $I_2 (\vec{a}', \vec{b}', \vec{c}') = \epsilon_{jkl}\, \det \rho\,a_j b_k c_l$ but how to extract the condition on $\epsilon_{imr}$?
$2)$ Show that these conditions are related to the canonical definition of $SO(3)$: $$SO(3) = \left\{O \in GL(3,\mathbb{R}) : O^TO = \mathbf{1} , \det O = 1\right\}$$
The conditions for $I_1$ and $I_2$ to be invariant are precisely that $\rho^T \rho = \mathbf{1}$ and that $\det \rho = 1$ from $1),$ so that makes me think what I did in $1)$ should been done in $2)$. If so, what are the conditions sought after in $1)$?
Thanks.