1. This universal property follows from the tensor algebra as a natural transformation. 2. This distinction is developed in greater detail in the article on tensor algebras . 3. Additional discussion of this point can be found in the tensor algebra article. 4. Here juxtaposition is symmetric respectively antisymmetric multiplication in the symmetric and antisymmetric tensor algebra . 5. The universal property can thus be seen as being inherited from the tensor algebra . 6. Thus, these two are consistency conditions for the Lie bracket on the tensor algebra . 7. The construction proceeds by first building the tensor algebra of the underlying vector space of the Lie algebra. 8. The monomials of these generators then generate the tensor algebra , on which the quotienting may be performed. 9. The result of the lifting is that the tensor algebra of a Lie algebra is a Poisson algebra. 10. The construction generalizes in straightforward manner to the tensor algebra of any " commutative " ring.
