(I found out the solution anyway so here it is)
The aforementioned householder equation also suppose that your plan contains the origin. So we cannot directly apply to your solution.
However let consider P as the new origin, x
coordinates in this system are now x - P
and the plan goes by the origin, its normal remains the same so you can compute S the symetrie of x in this system thanks to the HouseHolder equation:
S' = (x-p) - 2v(v^h(x-p)).
and you can get its coordinate in the original system :
S(x) = (x-p) - 2v(v^h(x-p)) + p
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