Let $F_n$ be the free group of rank $n\geq2$ with basis $x_1,\ldots,x_n$. For $1\leq j<i\leq n$, let $d_{i,j}$ and $e_{i,j}$ be the automorphisms of $F_n$ defined by $d_{i,j}(x_i)=x_ix_j$ and $e_{i,j}(x_i)=x_jx_i$, respectively, and fixing the remaining free generators. Write $D_n=\langle d_{i,j}\mid 1\leq j<i\leq n\rangle$ and $A_n^+=\langle d_{i,j},e_{i,j}\mid 1\leq j<i\leq n\rangle$. We prove that $D_n$ admits a presentation on the generators $d_{r+1,r}$, $1\leq r\leq n-1$, with three families of relations given by commutators of weights two, three, and four, respectively. We extend this to a presentation of $A_n^+$ on the generators $d_{r+1,r}$ and $e_{r+1,r}$, $1\leq r\leq n-1$. These generating sets have minimum cardinality. Moreover, the presentation of $A_n^+$ may be chosen so that every defining relator is a single commutator.
Let $F_n$ be a free group of rank $n\geq3$, freely generated by $x_1,\ldots,x_n$. For $1\leq j<i\leq n$, let $d_{i,j}$ denote the Nielsen automorphism of $F_n$ defined by $d_{i,j}(x_i)=x_ix_j$ and $d_{i,j}(x_k)=x_k$ for $k\neq i$, and let $D_n=\langle d_{i,j}\mid 1\leq j<i\leq n\rangle$. We determine the lower central series of $D_n$. We first prove that, for $1\leq r<i\leq n$, $d_{i,r}\in \gamma_{i-r}(D_n)\setminus\gamma_{i-r+1}(D_n)$. For each $i=2,\ldots,n$, this calculation leads to a filtration $\{W_{i,c}\}_{c\geq1}$ of $U_i=\langle d_{i,1},\ldots,d_{i,i-1}\rangle$. For every $c\geq1$, we obtain an explicit iterated semidirect-product decomposition of $\gamma_c(D_n)$ in terms of the subgroups $W_{i,c}$, and prove that $U_i\cap\gamma_c(D_n)=W_{i,c}$ for $i=2,\ldots,n$. The construction also gives an explicit basis for each quotient $\gamma_c(D_n)/\gamma_{c+1}(D_n)$ in terms of basic commutators and determines the exact lower-central depth of every such commutator. Consequently, $D_n$ is a Magnus group.