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 $n$, 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 $\Sigma_{n+1}$ (for $n>1$). 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.

I should probably explain my motivation, but unfortunately it is related to some preliminary results that have not been checked completely yet. I can only say that such structures appear on homogeneous pieces with respect to gradings of semisimple Lie algebras induced by nilpotent elements, and (somewhat enigmatically, I must admit) that all this is closely related to the answer of Noam D. Elkies to my previous question Seeking a more symmetric realization of a configuration of 10 planes, 25 lines and 15 points in projective space.