ONE-STEP BRACKET GENERATION AND EXTERIOR SQUARES: Classical Lie algebras, direct sums and packings

This article introduces and studies the bracket-generation number of a Lie algebra, defined as the minimal dimension of a subspace whose first-level brackets generate the entire algebra. The elementary combinatorial bound states that the binomial coefficient of this dimension must be greater than or equal to the dimension of the algebra. We prove that the classical families of simple Lie algebras of binomial dimension (odd-dimensional orthogonal and symplectic algebras) exactly attain this bound, with the sole exception of the orthogonal algebra of dimension four. Explicit witnesses with integer coefficients are provided, constructed via a complex structure and the moment curve. Furthermore, a structural connection is established between this invariant and a subspace packing problem involving direct exterior squares, proving a general overlap obstruction. We exactly solve the packing cases for subspaces of dimensions three and four.The exceptional case of dimension four is addressed using the Jacobi algebra, demonstrating that although no simple algebras with a square witness exist in this dimension, this non-semisimple algebra allows solving the problem and obtaining exact results for its direct sums.Finally, a general conjecture for the packing problem is formulated, the results are compared with other notions such as generation rank and Lie-length, and the cases of linear algebras of binomial dimension (such as solutions to the Pell equation) and exceptional algebras are left open.