[Verse 1] In the algebra's heart lies a maximal space Cartan subalgebra, abelian embrace Semisimple elements dance in formation Simultaneously diagonal across the nation H sits inside g like a compass true Pointing directions for what we can do [Chorus] Root systems classify, Dynkin draws the map A-B-C-D in classical wrap G-two F-four, E-six seven eight Exceptional beauties that mathematics create Decompose and conquer, let the structure show Root by root by root, watch the algebra grow [Verse 2] G equals H direct sum over all alpha phi G-alpha eigenspaces reaching for the sky Bracket H with X gives alpha H times X Linear functionals weaving geometric tricks Phi contains the roots in dual space they live Constrained configurations, only certain forms survive [Chorus] Root systems classify, Dynkin draws the map A-B-C-D in classical wrap G-two F-four, E-six seven eight Exceptional beauties that mathematics create Decompose and conquer, let the structure show Root by root by root, watch the algebra grow [Bridge] A-n special linear, circles on a line B-n orthogonal odd, double arrow shrine C-n symplectic forms, reverse arrow calls D-n orthogonal even, forked at branching walls [Verse 3] E-eight has dimension two-four-eight complete Two hundred forty roots in geometric feat String theory whispers secrets through its frame Periodic table of symmetry's grand game Nature chose these patterns, no others will do Mathematical surprises waiting to break through [Outro] From axioms simple, only finite types emerge Bracket laws and Jacobi make the structures converge Classical families guard the matrix throne While exceptional wonders stand beautifully alone
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