Given a set of weights of a representation and the corresponding extended Dynkin diagram for some Lie algebra, we can delete a node, which yields the maximal subalgebra. I know how to draw the extended Dynkin and that we add minus the highest root as a new node. Then we can compute the coefficient for every weight corresponding to this new root.
My problem is: Where do I insert this new coefficient? At the end? At the position of the new node in the Dynkin diagram? At the position of the node that I delete from the Dynkin diagram?
As an example let's consider $E_8$ and say we delete the third node, which means $E_8 \rightarrow SU(2) \otimes SU(3)\otimes A_5$. Where do I put the new coefficient in order to get the weights for the maximal subalgebra? Between 7 and 8, which is the position of the new node? Or at position 3, which would mean that I replace the deleted node/coefficient with the new node/coefficient?
I'm using the numbering and weight convention of LieArt. For the extended Dynkin diagram, see page 37 here
EDIT: It seems as this question is too trivial. Nevertheless a short one sentence comment would help me so much!
The coefficent is inserted at the position of the deleted simple root.
An explicit example can be found at page 73 here.
In addition at page 147 here Cahn writes, "... and use it in place of one of the old coefficents".