Systematic-matrix proof of extended Golay distance

Suppose a binary symmetric matrix $A$ satisfies $A^2=I$, has one row of Hamming weight eleven and eleven rows of Hamming weight seven, and every sum of two distinct rows has Hamming weight six. Then $(I\mid A)$ generates a self-dual code. Its rows have weights twelve or eight; their mutual orthogonality makes every generated word doubly even. A weight-four word $(u,uA)$ would have left-half weight one, two, three or four. The first two cases are excluded by the stated row weights; weight three would force $uA$ to be a unit vector and hence $u$ to be a row of $A$; weight four would force $uA=0$. Thus weight four is impossible and the minimum Hamming distance is eight.