Let \(K\) be a field and \(G\) be a subgroup of order 4 of the special linear group \(SL_2(K)\) generated by the matrix \(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\). Assume further that \(W\) is the associative unital algebra generated by two generic traceless matrices \(X\) and \(Y\); and \(L\) is the Lie subalgebra of the algebra \(W\) consisting of its Lie elements. If \(C(W)\) denotes the centre of \(W\), then the authors of the article determine the generators of the algebras \(W^G\) and \(L^G\) when considered as \(C(W)^G\)-modules. \(W^G\), \(L^G\) and \(C(W)^G\) denote the algebras of \(G\)-invariants of algebras \(W\), \(L\) and \(C(W)\) respectively. In particular, they show that \(W^G\) is freely generated by the elements \(I\) and \([X, Y]\), whereas \(L^G\) is generated by the element \([X, Y]\).