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ad(l algebra over F algebra Q algebra with involution arbitrary rings associative subalgebra canonical images characteristic zero commutative domain commutative ring contains a subalgebra countable number domain with identity eater-ideal epimorphism exists a unique following theorem free associative algebra free Lie algebra Hall Basis Theorem Heffalump algebra Hence ideal of Q identity of characteristic induction intersecting 2 trivially isomorphic Jordan algebras left—normal form Lie algebra homomorphism Lie polynomial Lie subalgebra linear Lie algebras linearly independent Lp(X LR(X monomials n—tad nilpotent elements non—Lie polynomial nonzero formal ideals nonzero ideal p(xp prime ideal Proof quotient algebra quotient field R-Id R—basis radical scalar-simple set of indeterminates simple Lie algebra skew Lie algebras skew(A special Lie algebra special linear Lie subalgebra free suffice to show Suppose symmetric indeterminate entries T-ideal theorems 2.9 triad products triad-eaters universal A—Heffalump algebra universal enveloping algebra vanishes identically