Let \(V\) be a linear space of finite dimension \(n\). Let lower case \(x_i\) denote elements of \(V\), which we will call also letters. We define a bracket as an alternating multilinear scalar valued function
\[
\begin {aligned}
[, \ldots , .] & : V \times \ldots \times V \rightarrow \mathbb {k} \quad (n\text {-factors}) \\
{\left [x_1, \ldots , x_n\right ]} & =\operatorname {sign}(p)\left [x_{p(1)}, \ldots , x_{p(n)}\right ] \\
{\left [x_1, \ldots , \alpha x_r+\beta y_r, \ldots , x_n\right ] } & =\alpha \left [x_1, \ldots , x_r, \ldots , x_n\right ]+\beta \left [x_1, \ldots , y_r, \ldots , x_n\right ]
\end {aligned}
\]
The sign is due to the permutation \(p\) on the arguments of the bracket. The pair \(\mathcal {P}=(V,[., \ldots ,])\). is called a Peano space.