I am trying to find an example of an algebra over a field of characteristic p (prime) which satisfies anti-symmetry and Jacobi identity but is not a lie algebra. i.e., [x,x] is not zero. 


Can one provide a pattern or a general method to modify existing Lie algebras in order to save anti-symmetry and Jacobi but **not** $[x,x]=0$ ?



Lie Algebra, counterexample
You must have $p=2$ otherwise antisymmetry implies the alternate identity $[x,x]=0$. 


The minimal example taylored on the pattern below I can provide is the (multiplicative) subalgebra (over the field $\mathbb{Z}/2\mathbb{Z}$) generated by
$$
E\_{12}+E\_{23}=
\begin{pmatrix}
0 & 1 & 0\\
0 & 0 & 1\\
0 & 0 & 0
\end{pmatrix}\ ;\ 
E\_{13}=
\begin{pmatrix}
0 & 0 & 1\\
0 & 0 & 0\\
0 & 0 & 0
\end{pmatrix}
$$  

then, you can check you have your algebra.


Now, if, for some reason, you need to preserve some identities or relations of an existing Lie algebra (always in characteristic $2$ for the reason said before), proceed as follows 


1. take you Lie algebra $(L,[-,-]\_L)$ over $k$ (of characteristic $2$) 
- add two dimensions by forming $L^{(1)}=L\oplus k.e\_1\oplus k.e\_2$ 
- extend the bracket $[x,y]$ to $L^{(1)}$ by
	1. $[x,y]\_L$ if $x,y\in L$
	- $[x,e\_i]=[e\_i,x]=0$ if $x\in L$
	- $[e\_1,e\_1]=e\_2;\ [e\_2,e\_i]=[e\_i,e\_2]=0$


then $L^{(1)}$ admits $L$ as a sector (direct summand and quotient), fulfills anti-symmetry and Jacobi, but not the alternate identity ($[x,x]=0$). 


