Lie superalgebra over the complex numbers

A complex Lie superalgebra is a complex vector space $\mathfrak{g}$ with a specified direct-sum grading $\mathfrak{g}={\mathfrak{g}}_{\bar{0}}\oplus {\mathfrak{g}}_{\bar{1}}$ and a complex-bilinear bracket $[,]:\mathfrak{g}\times \mathfrak{g}\to \mathfrak{g}$ satisfying the following conditions. A vector in one summand is homogeneous; its parity $\mathrm{\mid }x\mathrm{\mid }\in \{0,1\}$ is defined by $x\in {\mathfrak{g}}_{\overline{\mathrm{\mid }x\mathrm{\mid }}}$. For homogeneous $x,y,z$:

1. The bracket is graded: $[{\mathfrak{g}}_{\bar{i}},{\mathfrak{g}}_{\bar{j}}]\subseteq {\mathfrak{g}}_{\overline{i+j}}$.
2. It is graded-skew: $[x,y]=-(-1{)}^{\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }y\mathrm{\mid }}[y,x]$.
3. It satisfies the signed Jacobi identity in derivation form: $[x,[y,z]]=[[x,y],z]+(-1{)}^{\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }y\mathrm{\mid }}[y,[x,z]]\mathrm{.}$
4. The odd self-Jacobi identity $[x,[x,x]]=0$ for odd $x$ is listed explicitly as part of this convention. Over $\mathbb{C}$ it follows from condition 3.

Here $\mathfrak{g}$ is a vector space in the sense of Vector space over a field, and the bracket is bilinear in the sense of Linear map between vector spaces over the same field. The ground field is fixed to $\mathbb{C}$ throughout this pair; no positive-characteristic claim is included.

The even part ${\mathfrak{g}}_{\bar{0}}$ is an ordinary Lie algebra under the restricted bracket. Odd brackets are symmetric, and may be nonzero on the diagonal: if $x$ is odd, graded skew symmetry places no condition forcing $[x,x]=0$. The extra odd identity is redundant in characteristic zero; in characteristic three it must be imposed separately for the usual PBW route.