Prove that if the Lie group \(G_{1}\), with generators \(J_{i}(1)\), is isomorphic to a group \(G_{2}\), with generators \(J_{i}(2)\), and \(J_{i}(1)\) and \(J_{i}(2)\) operate on independent degrees of freedom, then the sum operators \(J_{i} \equiv J_{i}(1)+J_{i}(2)\) obey the same Lie algebra as the generators of \(G_{1}\) and \(G_{2}\). Thus the \(J_{i}\) generate a group \(G\) that is isomorphic to \(G_{1}\) and \(G_{2}\).