Diagrammatic technique for Vogel's universality

In his 1999 preprint "Universal Lie Algebra", P. Vogel put forward a hypothesis on the existence of a universal Lie algebra. Although this hypothesis remains open, it is known that many quantities in Lie theory admit universal descriptions. Remarkably, almost all such universal formulas have been obtained through the representation theory of simple Lie (super)algebras, whereas Vogel's original framework was based on a more abstract diagrammatic algebra. Nevertheless, the diagrammatic approach has received little attention over the past two decades, since the last contributions by P. Vogel and J. Kneissler. In this work, we revive the diagrammatic technique grounded in Vogel's $\Lambda$-algebra and show that it enables truly universal computations. We examine numerous examples and discuss them.