Convert local coordinate system to global like WGS84

Question

I have created a local coordinate system with several points and I am currently trying to convert this local system to a global one with the aid of the Helmholtz transformation. I got a solution with the github code of this thread https://gis.stackexchange.com/questions/386393/convert-x-y-z-local-points-to-wgs84-having-only-3-coordinates-known. However, the used code is hard to understand and I was wondering how I could transform my COS more easily? I have three local points which are linked to global ones like this:

# local cartesian [x, y, z] Control points
local_ctrl_p = [[0, 0, 0],
                [3940, 0, 0],
                [0,1000,0]]

# geodetic [lat,lon,h] Control points
geodet_ctrl_p = [[50.045105,8.586917, 108],
                 [50.032637,8.534725, 97],
                 [50.053528490907745, 8.58205606668748 , 113.1159057617188]]

How am I able to transform the following points more easily?

local_other_p = [[0, 1000, 0],
                 [0, -1000, 0],
                 [1000, 0, 0],
                 [-1000, 0, 50]]