We give a unified division algebraic description of (D = 3, $$ \mathcal{N} $$ = 1, 2, 4, 8), (D = 4, $$ \mathcal{N} $$ = 1, 2, 4), (D = 6, $$ \mathcal{N} $$ = 1, 2) and (D = 10, $$ \mathcal{N} $$ = 1) super Yang-Mills theories. A given (D = n + 2, $$ \mathcal{N} $$ ) theory is completely specified by selecting a pair ( $$ {\mathbb{A}}_n $$ , $$ {{\mathbb{A}}_n}_{\mathcal{N}} $$ ) of division algebras, $$ {\mathbb{A}}_n $$ ⊆ $$ {{\mathbb{A}}_n}_{\mathcal{N}} $$ = $$ \mathbb{R},\mathbb{C},\mathrm{\mathbb{H}},\mathbb{O} $$ , where the subscripts denote the dimension of the algebras. We present a master Lagrangian, defined over $$ {{\mathbb{A}}_n}_{\mathcal{N}} $$ -valued fields, which encapsulates all cases. Each possibility is obtained from the unique ( $$ \mathbb{O},\mathbb{O} $$ ) (D = 10, $$ \mathcal{N} $$ = 1) theory by a combination of Cayley-Dickson halving, which amounts to dimensional reduction, and removing points, lines and quadrangles of the Fano plane, which amounts to consistent truncation. The so-called triality algebras associated with the division algebras allow for a novel formula for the overall (spacetime plus internal) symmetries of the on-shell degrees of freedom of the theories. We use imaginary $$ {{\mathbb{A}}_n}_{\mathcal{N}} $$ -valued auxiliary fields to close the non-maximal supersymmetry algebra off-shell. The failure to close for maximally supersymmetric theories is attributed directly to the non-associativity of the octonions.