B[a_, b_, c_, d_, e_, 
   f_] = {{0, -a, -b, -d}, {a, 0, -c, -e}, {b, c, 0, -f}, {d, e, f, 
    0}};
B1 = B[1, b, c, 0, 0, 0];
B2 = B[0, 0, 0, d, e, f];
Solve[{Dot[B1, B2] == Dot[B2, B1]}, {b, c, d, e, f}]