i use brass as material.

measure the diameter of the head stock hole.

Make your product
0.1mm smaller

Drill a hole of 12.1mm

Make 2 centers in the middle and drill them 3.3mm for m4 ( do not make the screw threat yet.

cut your product with an angle in to two pieces. You can do this also with a saw.

drill one piece bigger with
4.3mm

in the other you tapped m4

in the upper piece you make a small screw to fix the ax

put it in the head stock of your lathe.

turn on the screws. You shall see that with a little strength your tool is fixed.

now it’s ready to use.