3 Point Align

Hello,

in an AR-Application I would like to align loaded scanned realworld meshes, with the current AR View of the real world.

The user should cklick 3 points on the scanned meshes (source), afterwards users should click 3 corresponding points in realworld (target). Therefore I have 2 sets, each with 3x Vector3 and want them to be used to calculate the transform to align the scanned meshes on realworld accordingly.

I tried several things but can not get right mathematic to get the desired reuslt:

-take each set of three points as a "plane", get those plane normals and use Quternion.LookRotation makes the "planes" parallel, but not in correct rotation (tried flipped or not flipped in various combinations)

 private void StartAlign(List<Vector3> src, List<Vector3> trg)
 {
         PlaneSrc = new Plane(src[0], src[1], src[2]);
         PlaneTrg = new Plane(trg[0], trg[1], trg[2]);
 
     Object_Scanned.transform.rotation = Quaternion.LookRotation(PlaneTrg.flipped.normal, PlaneSrc.normal);
 }

-get the matrix4x4 for each set, calculate and attach, but none of my mathematic tries brought up what i need (below just one example fpr matrix4x4):

 public void Align3Point(GameObject Objects_Scanned, GameObject to_target, List<Vector3> align_points, List<Vector3> target_points)
 {
         Matrix4x4 sourcePlane = MatrixFrom3Points(align_points[0], align_points[1], align_points[2]);
         Matrix4x4 targetPlane = MatrixFrom3Points(target_points[0], target_points[1], target_points[2]);
 
         alignPlane = Matrix4x4.Rotate(Quaternion.Lerp(sourcePlane.rotation, targetPlane.rotation, 360f));
 
         Objects_Scanned.transform.rotation = Quaternion.LookRotation(alignPlane.GetColumn(2), alignPlane.GetColumn(1));
 }
 
 public Matrix4x4 MatrixFrom3Points(Vector3 p0, Vector3 p1, Vector3 p2)
 {
         Vector3 x = p1 - p0;
         x.Normalize();
         Vector3 y = p2 - p0;
         Vector3 z = Vector3.Cross(x, y);
         z.Normalize();
         y = Vector3.Cross(z, x);
         y.Normalize();
 
         return new Matrix4x4(new Vector4(x.x, x.y, x.z, 0),
                              new Vector4(y.x, y.y, y.z, 0),
                              new Vector4(z.x, z.y, z.z, 0),
                              new Vector4(p0.x, p0.y, p0.z, 1));
 }

Can someone help me with the mathematic behind a 3 point align please?

Thank you so much