[Verse 1] Take a vector space and add a bracket operation Bilinear and smooth, that's our foundation X with X equals zero, alternating they say Jacobi identity keeps chaos at bay [Chorus] Lie algebras, structure so divine Brackets and ideals, everything's aligned Simple or semisimple, we can classify With the Killing form, non-degenerate why [Verse 2] Ideals are subspaces with a special trait Bracket with the whole space, they stay inside the gate Like normal subgroups but in this algebra land They help us understand structure, so grand [Chorus] Lie algebras, structure so divine Brackets and ideals, everything's aligned Simple or semisimple, we can classify With the Killing form, non-degenerate why [Bridge] Simple means non-abelian, no proper ideals inside Semisimple is the sum where simple algebras reside Cartan's criterion tells us what we need to know Killing form non-degenerate, that's how the theorems go [Verse 3] Adjoint representation, ad of X takes Y To the bracket X with Y, that's how we define Killing form takes two elements, computes the trace Of adjoint compositions in this vector space [Chorus] Lie algebras, structure so divine Brackets and ideals, everything's aligned Simple or semisimple, we can classify With the Killing form, non-degenerate why [Outro] From alternating brackets to the Killing trace Lie algebra structure has its rightful place Remember Jacobi, remember what's inside Mathematical beauty we cannot hide
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