We have presented here methods which illustrate how root and weight diagrams
can be used to visually identify the subalgebras of a given Lie algebra.
While the standard methods of determining subalgebras rely upon adding,
removing, or folding along nodes in a Dynkin diagram, we show here how to
construct any of a Lie algebra's root or weight diagrams from its Dynkin
diagram, and how to use geometric transformations to visually identify
subalgebras using those weight and root diagrams. In particular, we show how
these methods can be applied to algebras whose root and weight diagrams have
dimensions four or greater. In addition to pointing out the erroneous
inclusion of $C_4 \subset F_4$ in [15, 16], we provide
visual proof that $C_4 \subset E_6$ and list all the subalgebras of $E_6$.
While we are primarily concerned with the subalgebras of $E_6$, these methods
can be used to find subalgebras of any rank $l$ algebra.
6. References
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