It is well known that the standard bracketings of Lyndon words in an alphabetAAform a basis for the free Lie algebraLie(A){\text {Lie}}(A)generated byAA. Suppose thatg≅Lie(A)/J\mathfrak {g} \cong {\text {Lie}}(A)/Jis a Lie algebra given by a generating setAAand a Lie idealJJof relations. Using a Gröbner basis type approach we define a set of "standard" Lyndon words, a subset of the set Lyndon words, such that the standard bracketings of these words form a basis of the Lie algebrag\mathfrak {g}. We show that a similar approach to the universal enveloping algebrag\mathfrak {g}naturally leads to a Poincaré-Birkhoff-Witt type basis of the enveloping algebra ofg\mathfrak {g}. We prove that the standard words satisfy the property that any factor of a standard word is again standard. Given root tables, this property is nearly sufficient to determine the standard Lyndon words for the complex finite-dimensional simple Lie algebras. We give an inductive procedure for computing the standard Lyndon words and give a complete list of the standard Lyndon words for the complex finite-dimensional simple Lie algebras. These results were announced in [LR].

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