Combinatorial problem: triple binomial product related to squared Laguerre polynomials

Question

Context

Hydrogenic wavefunctions [1] include a factor given by Laguerre polynomials [2]. These wavefunctions are often encountered in a first course in quantum mechanics. They also appear in subsequent courses in atomic physics. These wavefunctions are also applicable to the study of two-electron atoms such as helium. In the course of my studies, I realized that I can not determine how to derive the identity below. I have checked that it is true using software, but I can not show it.

Question

Given $n\in\mathbb{N}$ and $\ell\in\left\{\mathbb{N}\mid n>\ell\geq 0\right\}$, how can the following identity be proven: \begin{align*} \frac{2 \,n \left(n + \ell\right)!} {(n - \ell - 1)!} &= \sum_{j=0}^{2\left[n - \ell - 1\right]} \frac{(-1)^{j} \left(2\,\ell+2+j\right)! }{j!} \sum_{k=0}^{n - \ell - 1 - \left|n - \ell - 1-j \right| } \\ &\qquad\qquad \times {n + \ell \choose k - \left[ j-n + \ell + 1 \right]\left[1-u(j - n + \ell + 1 ) \right]} \\ & \qquad\qquad\times {n + \ell \choose j - k - \left[j - n + \ell + 1\right]\left[1 +u(j - n + \ell + 1)\right] } \\ & \qquad\qquad\times {j \choose k + ( j - n + \ell + 1 )\,u(j - n + \ell + 1) } \end{align*} Above, $u$ is the Heaviside-step function.

My attempt

I looked in CRC and I looked online trying to find a form similar to what I have here. On [4] I found Dixon's identity, which does include a product of three binomial coefficients. However, Dixon's identity does not meet my needs. This is so since in my problem there is a factor $(-1)^j$, which does not appear in Dixon's identity. I have searched this site and found [5]. I found at least one identity that includes the product of two binomials (e.g., cf., [3]); however, I could not find any identities with the product of three binomials. I have looked on this site for products of three binomials. In [5], I find a question with three binomial coefficients whose problem statement includes several leads, which will be explored in due time. In [6], I also have found a question with a similar expression to mine and the question in [6] also includes a factor $(-1)^j$.

Bibliography

[1] https://en.wikipedia.org/wiki/Hydrogen_atom

[2] https://en.wikipedia.org/wiki/Laguerre_polynomials#Multiplication_theorems

[3] Proof of $\sum_{m=0}^{n}\binom{m}{j}\binom{n-m}{k-j}=\binom{n+1}{k+1}$ (Another form of the Chu–Vandermonde identity)

[4] https://en.wikipedia.org/wiki/Binomial_coefficient

[5] Finite sum with three binomial coefficients

[6] sum of product of three binomial coefficients