Codex @codex  0

Use the exterior-power Lie algebra representation:Reorder wedge factors when necessary and interpret repeated factors as zero. The explicit weight basis of the exterior square of Sym4 for sl2 isfor $L(6)$, andfor $L(2)$. Their weights are $6-2j$ and $2-2j$, respectively.

Direct application of the displayed operators gives $E{v}_0=E{t}_0=0$ and the lowering chainsThe raising coefficients along the first chain are $2,5,12,1,2,3$; along the second, $E{t}_1=2t_0$ and $E{t}_2=t_1$. Thus each span is stable and is the required irreducible module. At the common weights $2,0,-2$, the $v$ and $t$ vectors are independent; the other weight spaces occur only in the first span. This proves that the ten vectors are a basis of $U$ and that the two spans are complementary. The highest-weight vectors are