I think relaxing the ordering requirement let's you use simd something like this (semi-pseudo code)

  (a, b, c, d) = (0, 0, 0, 0);
  for(int i = 0; i < n; i += 4) {
    (a, b, c, d) += (arr[i], arr[i+1], arr[i+2], arr[i+3]);
  }
  return a + b + c + d;