Exercise 3, § 1

Let $M$ be a not necessarily associative algebra over a field $K$, $(e_i)_{i \in I}$ a basis of the vector space $M$ and $c_{ijk}$ the constants of structure of $M$ relative to the basis $(e_i)$. For $M$ to be a Lie algebra, it is necessary and sufficient that the $c_{ijk}$ satisfy the following conditions: (a) $c_{ilk} = 0$; (b) $c_{ijl} = -c_{jik}$; (c)