Frobenius Map on BN Curve sextic twist?
In step d) $Q_1$ is a point after applying the Frobenius map on $Q$. Then in step e) there is $T=T+Q_1$. Confusion: According to my calculation above, $Q_1$ should not be on the same curve where $T$ is. Is it correct to calculate $T=T+Q_1$? Or is my calculation wrong?


It looks like CurveEnthusiast verified your calculation in comments, so this is strictly for your final question:

The Frobenius map $\pi$ is a mapping from $\mathbb{E}′(\mathbb{F}_q)$ to itself. $T$ is a point on this curve, and since $\pi:\mathbb{E}′(\mathbb{F}_q) \rightarrow \mathbb{E}′(\mathbb{F}_q)$, $Q_1$ is also on the curve and so the sum is defined.