In Humphreys's book "Representations of semi-simple Lie algebras in the BGG-category O", section 3.14 deals with contravariant forms on highest weight modules. I wanted to define a map by a bilinear form and came across the following problem.

Let $V$ and $W$ be two irreducible $\mathfrak{g}$-modules, where $\mathfrak{g}$ is a semi-simple Lie algebra.

Then there exists a unique (up to scalar) contravariant non-degenerate form on $V$ and $W$ giving rise to a contravariant non-degenerate form on $V \otimes W$.

On the other hand, decomposing $V\otimes W$ into irreducible $\mathfrak{g}$-modules, we also have contravariant non-degenerate forms on each summand, yielding a contravariant non-degenerate form on the sum.

Is there any chance, that these two bilinear forms on $V \otimes W$ are the same?

$\rule{330pt}{0.4pt}$

$\textbf{Edit:}$ Here is my special case of the above question, which may be easier to understand. Let $\mathfrak{g}$ be a simple Lie algebra and let $\theta$ be the highest root, such that $\mathfrak{g} = V(\theta)$ as irreducible $\mathfrak{g}$-modules.

$V(k\theta)$ is a submodule of $\mathfrak{g}^{\otimes k}$ by sending the highest weight vector to $e_\theta^{\otimes k}$. Hence also $V(k\theta) \otimes V(l\theta)$ can be viewed as a submodule of $\mathfrak{g}^{\otimes k+l}$. $V(k\theta) \otimes V(l\theta)$ admits a contraviariant bilinar form $\beta$ coming from the forms on $V(k\theta)$ and $V(l\theta)$. I would like to show, that the form on $\mathfrak{g}^{\otimes k+l}$ coming from the Killing form restricts to $\beta$ on $V(k\theta) \otimes V(l\theta)$.