Are edges of inclusion minimal connected supergraphs bases of a matroid?

Question

My reasoning is for any non-connected graph $G=(V,E)$ we can define an equivalence $\sim$ on $V$ such that for all $a,b\in V$ we get $a\sim b$ iff there is an undirected walk from $a$ to $b$ in the graph $G$ then $M=(E\cup\{\{u,v\}:(u,v)\in \cup_{(A,B)\in V/\sim:A\neq B}A\times B\},\mathcal{I})$ would be a matroid such that $I\in\mathcal{I}\iff\forall \{u_1,v_1\},\{u_2,v_2\}\in I\setminus E(u_1\sim u_2\implies \{u_1,v_1\}=\{u_2,v_2\})$, thus if we define the graph $H=(V,E\cup\{\{u,v\}:(u,v)\in \cup_{(A,B)\in V/\sim:A\neq B}A\times B\})$ and let $\mathcal{T}$ be the edges of the tree graphs spanning $H$ that contain a spanning tree of $G$ then the bases of $M$ are $\{E\cup T:T\in\mathcal{T}\}$ thus we can form $M$ by adding coloops to the graphic matroid of $H$.... correct?