A four vector $a_i$ transforms in $SO(4)$ as $\underline{4} \sim(\underline{2},\underline{2})$ under the algebra $so(4)\sim su(2)\times su(2)$. How does a symmetric tensor $a_i b_j + a_j b_i$ transform? (under $so(4)$))
A symmetric tensor has 10 independent entries, so I need to understand how a 10-dimensional representation decomposes in $su(2)\times su(2)$ and if it is irreducible I think...
Symmetric tensors are not irrecducible, they decompose into tracefree tensors (in your notation, this means $g^{ij}a_ib_j=0$ and a trace part, which must be a multiple of $g_{ij}$. Thus you get a direct sum of a trivial representation and a nine-dimensional one, which is just $(\underline{3},\underline{3})$.