I would like to figure out whether there is an irreducible complex (in the sense non-self-conjugate) representation of a group $SO(N)$, $N>2$, whose tensor square contains the fundamental representation.

$SO(N)$ only has complex representations for $N$ even, $N\geq 6$. When one allows projective representations, then the spinor representation of the algebra $so(6)=sl(4)$ does the job, but I am interested only in non-projective representations of the orthogonal group.

I think a higest-weight representation $\Gamma_{a_1,a_2,a_3}$ of $sl(4)$ gives a representation of $SO(6)$ if and only if $a_1+a_3$ is even, so that the total number of boxes in the corresponding Young tableau is $2(a_1+a_2)+a_1+a_3$, which is even. It follows from the Littlewood-Richardson rule that the tensor square of $\Gamma_{a_1,a_2,a_3}$ only contains irreps whose Young tableaux have $4k$ boxes, so can't contain the fundamental of $SO(6)$, i.e. $\Gamma_{0,1,0}$. Hence a possible example must have $N\geq 8$.

Any ideas?