Suppose $V_\lambda$ and $V_\mu$ are two irreducible Lie Algebra representations with highest weight vectors $v_\lambda$ and $v_\mu$, respectively. I can decompose the tensor product $V_\lambda \otimes V_\mu = \oplus V_{\nu}^{m_\nu}$ and I would like to know a systematic way to figure out the highest weight vectors of each $V_\nu$ in terms of the highest weight vectors $v_\lambda$ and $v_\mu$.
For a simplest example that I am working with for intuition, I have $V_\lambda = V_\mu = \mathfrak{sl}_n$, the adjoint representation. For simplicity of notation let $n= 6$, then I have that $[1 \ 0 \ 0\ 0\ 1] \otimes [1 \ 0\ 0\ 0\ 1] = [2\ 0\ 0\ 0\ 2] \oplus [2\ 0\ 0\ 1\ 0] \oplus [ 0\ 1\ 0\ 0\ 2] \oplus [0\ 1\ 0\ 1\ 0] \oplus 2[ 1\ 0\ 0\ 0\ 1] \oplus [ 0\ 0\ 0\ 0\ 0]$.
I know that $v_\lambda \otimes v_\mu= v_{\lambda + \mu}$, where $v_\lambda = E_1^n$ and $v_\mu = F_1^n$, would give me the highest weight vector of $[2\ 0\ 0\ 0\ 2]$, but I am unsure how to obtain highest weight vectors of the other representations in the decomposition in a systematic fashion.
I understand that this problem in full generality is quite difficult, but I'd appreciate any guidance or perhaps a reference where I can understand this. Please let me know if anything requires further clarification. Thank you for your time!