Let's start with a definition of an $n$-braid, as I understand them:
Definition: Let $n$ be a natural number. In $ \mathbb{R}^2$, consider two collections of $n$ nodes $\{P_i\}$ and $\{Q_i\}$. Let the $P_i$ lie at the points $(x,y)=(0,i)$ and the $Q_i$ lie at $(a,i)$ for some $a\in\mathbb{R}^+$. Consider $\mathbb{R}^2$ to exist as a plane within $\mathbb{R}^3$. An $n$-braid is a collection of $n$ non-intersecting, finitely interlaced strands that start on the nodes of $\{P_i\}$ and move in the strictly positive $x$-direction toward the nodes of $\{Q_i\}$. A node is either the destination or beginning of exactly one strand.
Consider $\sigma_i$ to be the swapping of the $(i+1)$th strand over the $i$th:
Similarly, consider $\sigma_i^{-1}$ to be the swapping of the $i$th strand over the $(i+1)$th, so that $\sigma_i\sigma_i^{-1}$ is the identity braid. Assume composition is from left to right. How would we show that any $n$-braid is a composition of braid transpositions? An example being the rather complex pure ($P_i\mapsto Q_i$) 4-braid below, which is decomposed using the relation $\sigma_i\sigma_j=\sigma_j\sigma_i \iff |i-j|>1$:
This braid would then have braid word $\sigma_1^{-1}\sigma_2\sigma_3^{-1}\sigma_2^{-1}\sigma_3\sigma_1^{-1}\sigma_2\sigma_1^{-1}\sigma_2\sigma_3^{-1}$:
Any ideas about how one might go about proving that any $n$-braid is a product of transpositions? We can assume both relations hold, including $\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}$. I am rather against hindering the clarity of such a proof by relying on heavy calculus, so I would be most appreciative if someone could share their insight on how one might operate within the realm of algebra.
The result is due to Artin. You can find a proof for instance in
Birman, Joan S., Braids, links, and mapping class groups. Based on lecture notes by James Cannon, Annals of Mathematics Studies. 82. Princeton, N. J.: Princeton University Press and University of Tokyo Press. IX, 229 p. (1975). ZBL0305.57013.
Specifically, Theorem 1.8 which might to your taste since it is mostly algebraic.
Or, if you prefer, read a proof in section 5.3 of
Prasolov, V. V.; Sossinsky, A. B., Knots, links, braids and 3-manifolds. An introduction to the new invariants in low-dimensional topology. Transl. by A. B. Sossinsky from an orig. Russian manuscript, Translations of Mathematical Monographs. 154. Providence, RI: American Mathematical Society (AMS). viii, 239 p. (1997). ZBL0864.57002.
Their proof is very geometric and intuitive, so maybe you do not like this.