I am trying to reformulate the constraints
$$ \alpha^\intercal L \beta + \|L^\intercal \alpha\|_{2}^{2} \leq \rho, $$
where $\alpha\in\mathbb{R}^{n},\beta\in\mathbb{R}^{m}$ and $\rho\in\mathbb{R}$ are constants, and $L\in\mathbb{R}^{n\times m}$ is the decision variable of interest. These constraints are quadratic constraints, and I believe they are convex because I know that $L$ has full row rank.
However, I am having trouble formulating these constraints in a more standard/canonical form, such as second-order cone constraints (SOCC). My main issue is with the linear term $\alpha^\intercal L\beta$ that does not split up nicely as $\|z\|_2^2$, for example. Any help or insights on these constraints would be incredibly helpful!