Solution 2

x1 = [2.875, -3.875, -0.625, 1.625, -0.7010, -0.7010, 0.1305, 0.0000, 0.0239, 0.0127, 0.7104, 0.7032] # 0.01614

qR={p,q,r,s}={0.023910908577807,0.012725925514069,0.710444323632657,0.703231954826379};
qL={a,b,c,d}={-0.701057384649978,-0.701057384649978, 0.130526192220052,0};
ML=(({{a, -b, -c, -d}, {b, a, -d, c}, {c, d, a, -b}, {d, -c, b, a}}));
MR=(({{p, -q, -r, -s}, {q, p, s, -r}, {r, -s, p, q}, {s, r, -q, p}}));
ML.MR//MatrixForm