simple contact algebra

Question

Let suppose we have contact lie algebra K(3,(2,2,2)) over GF(3), according to the book "Modular lie algebras and their representations" from Helmut Strade, K(3,(2,2,2)) is simple Lie algebra ( it means that it has no ideal except trivial and itself) and its dimension is equal to $729$. I want to investigate K(3,(2,2,2)) over GF(2), How can we check that it simple or not? what is its dimension ( 64 or 63) ? If it is not simple how can we find nilpotent radical of K(3,(2,2,2))?

Answer 1

We have $K(m,\underline{n})=K(m,\underline{n})^{(1)}$ unless $m\equiv -3\mod p$. But for $m=3$ and $p=2$ this congruence holds. So the algebra $K(3,\underline{n})$ over $GF(2)$ is not simple. In general, the derived algebra $K(m,\underline{n})^{(1)}$ is simple. Its dimension is $p^{\sum_in_i}-1$ if $m\equiv -3 \mod p$, and $p^{\sum_in_i}$ otherwise. For $m=3$ and $p=2$ we obtain that $\dim K(3,(2,2,2))^{(1)}=2^{2+2+2}-1=63$.