I have seen two kinds of definitions of irreducible connections on fibre bundls: A connection is said to be irreducible if

- the holonomy group is precisely $G$ and not a proper subgroup.

or 2. there exist a constant $C$ depending on the connection $D$ such that \begin{equation} \|s\|_{H^{1,2}} \le C\|Ds\|_{L^2} \end{equation} for all $s\in H^{1,2}(\Omega^0(ad\eta))$ .

My question is: what is the relation and difference of the above two definitions?