Proposition 3

Let $K'$ be a non-zero commutative ring and $u : K \to K'$ a ring homomorphism. For every set X there exists one and only one Lie $K'$-algebra homomorphism

such that $v(x \otimes 1) = x$ for $x \in X$. Further, v is an isomorphism.

Applying Proposition 1 to $g = L_{K'}(X)$ considered as a Lie K-algebra and the mapping $x \mapsto x$ of X into g, we obtain a K-homomorphism $L_K(X) \to L_{K'}(X)$, whence there is a $K'$-homomorphism $v : L_K(X) \otimes K' \to L_{K'}(X)$. The fact that v is unique and is an isomorphism follows from the fact that the ordered pair $(L_K(X) \otimes K', x \mapsto x \otimes 1)$ is a solution of the same universal mapping problem as the ordered pair $(L_{K'}(X), x \mapsto x)$.