Lie superalgebra

Remarks

Example: let $V = V_{\bar 0} \oplus V_{\bar 1}$ with $V_{\bar 0} = k^m$ and $V_{\bar 1} = k^n$. The space $\mathfrak{gl}(m|n) = \End_k(V)$, graded by the maps that preserve or swap the two parts, with $[x, y] = xy - (-1)^{\abs{x}\abs{y}} yx$ on homogeneous elements, is a Lie superalgebra. Clause 1 holds because parities add under composition, and clauses 2 and 3 follow from the associativity of composition by expanding the signed products.

Non-example: take the same graded space $\mathfrak{gl}(1|1)$ but the ordinary commutator $[x, y] = xy - yx$. For the odd elements $x = E_{12}$ and $y = E_{21}$, clause 2 requires $[x, y] = [y, x]$, but $xy - yx = E_{11} - E_{22}$ while $yx - xy = E_{22} - E_{11}$. So clause 2 fails.

The even part $\mathfrak{g}_{\bar 0}$ is an ordinary Lie algebra, and the bracket makes $\mathfrak{g}_{\bar 1}$ a $\mathfrak{g}_{\bar 0}$-module. A Lie superalgebra with $\mathfrak{g}_{\bar 1} = 0$ is just a Lie algebra. Over fields of characteristic $3$ one adds the axiom $[y, [y, y]] = 0$ for odd $y$, and characteristic $2$ needs a separate treatment.