Pensieve header: Matrix implementation of Ig (failed).

B[x_,y_] := x.y-y.x;
M[a_,b_,c_,d_] := (({{a, b, 0, 0}, {0, 0, d, c}, {0, 0, 0, -b}, {0, 0, 0, -a}}));
MatrixForm /@ ({c,w,b,u}=M @@@ IdentityMatrix[4])

B[c,w] // MatrixForm

B[w,b] // MatrixForm

B[u,b] // MatrixForm

B[c,b] // MatrixForm

B[c,u] // MatrixForm

{B[c,w]==w}