Ordered monomial basis of a symmetric algebra

Let $P$ be the vector space freely spanned by the finite weakly increasing words in $B$, including the empty word. Multiply two basis words by sorting their concatenation. Totality of the supplied order makes the sorted word unique; this multiplication is commutative and associative, and the empty word is its unit.

By [L1], sending $b\in B$ to the one-letter word $b$ extends uniquely to a linear map $j:V\to P$. By [L2], it extends to a unital algebra map $\alpha :S(V)\to P$.

Send each ordered word $b_1\cdots b_n$ in the basis of $P$ to the product of the images of its letters in $S(V)$, and send the empty word to $1$. Extending linearly gives an algebra map $\beta :P\to S(V)$ because multiplication in $S(V)$ is commutative and the product of two word images is the image of their sorted concatenation.

The composite $\alpha \beta$ fixes every word-basis element of $P$, while $\beta \alpha$ fixes every generator from $V$ and hence all of $S(V)$ by [L2]. Thus $\alpha$ and $\beta$ are inverse isomorphisms.

Consequently the stated ordered words are a basis of $S(V)$. The construction uses only the supplied basis and order, and when $B$ is empty the empty word is the sole basis element.