The term “Dynkinese” is from the book Journeys Beyond The Standard Model by P. Ramond and describes the usage of Dynkin's methods for group theoretical computations. In practice this means, we work with weights instead of tensors. This is especially useful for complicated groups like $E_6$. With Dynkin's methods, we don't need to construct possibly complicated tensors, but can work instead with much simpler weight vectors. This works for any representation, no matter how large and for any group.
“every abstract Dynkin diagram comes from a complex simple Lie algebra” http://www.mat.univie.ac.at/~cap/files/wisser.pdf
See page 74 in this lectures for explanation for weights from extended Dynkin symmetry breaking. The new coeffiecent for minus the highest weight is added at the position of the deleted root.
This is also explained and demonstrated nicely in the LieArt documentation.