I have encountered very strange commutative nonassociative algebras without unit, over a characteristic zero field, and I cannot figure out where do they belong. Has anybody seen these animals in any context?

For each natural $ n>1$ , the $ n$ -dimensional algebra $ A_n$ with the basis $ x_1$ , …, $ x_n$ has the multiplication table $ x_i^2=x_i$ , $ i=1,…,n$ , and $ x_ix_j=x_jx_i=-\frac1{n-1}(x_i+x_j)$ (for $ i\ne j$ ).

The only thing I know about this algebra is its automorphism group, which is the symmetric group $ \Sigma_{n+1}$ . This can be seen from how I obtained the algebra in the first place.

Let $ I$ be the linear embedding of $ A_n$ onto the subspace of the $ (n+1)$ -dimensional space $ E_{n+1}$ of vectors with zero coefficient sums in the standard basis, given by $ $ I(x_i)=\frac{n+1}{n-1}e_i-\frac1{n-1}\sum_{j=0}^ne_j,\quad i=1,…,n $ $ and retract $ E_{n+1}$ back to $ A_n$ via the linear surjection $ P$ given by $ $ P(e_0)=-\frac{n-1}{n+1}(x_1+…+x_n) $ $ and $ $ P(e_i)=\frac{n-1}{n+1}x_i,\quad i=1,…,n. $ $ Then the multiplication in $ A_n$ is given by $ $ ab=P(I(a)I(b)), $ $ where the multiplication in $ E_{n+1}$ is just that of the product of $ n+1$ copies of the base field; more precisely, it has the multiplication table $ $ e_ie_j=\delta_{ij}e_i,\quad i,j=0,…,n $ $ (where $ \delta$ is the Kronecker symbol).

$ A_n$ is not Jordan either, in fact even $ (x^2)^2\ne(x^2x)x$ in general.

Really don’t know what to make of it.