Linear independence does not really say much.

This algebraic variety is discussed in some detail in an old paper of Kirillov and Neretin: The variety $A_n$ of $n$-dimensional Lie algebra structures.

The case of $n=4$ which already shows many interesting phenomena (but already is nontrivial from the computer algebra viewpoint) is analysed in detail in the recent preprint Manivel, Sturmfels, and Sverrisdóttir - Four-Dimensional Lie Algebras Revisited — you might find it enlightening.

Linear independence does not really say much.

This algebraic variety is discussed in some detail in an old paper of Kirillov and Neretin: The variety $A_n$ of $n$-dimensional Lie algebra structures.

The case of $n=4$ which already shows many interesting phenomena (but already is nontrivial from the computer algebra viewpoint) is analysed in detail in the recent preprint Manivel, Sturmfels, and Sverrisdóttir - Four-Dimensional Lie Algebras Revisited — you might find it enlightening.

Update: I checked the MathSciNet review of the paper of Kirillov and Neretin and found two other relevant references:

Carles, Diakité - les variétés d'Algèbres de Lie de dimension $\leqslant 7$

Gorbatsevich - Some properties of the space of n-dimensional Lie algebras (where in particular your observation on linear independence is proved)

Linear independence does not really say much.

This algebraic variety is discussed in some detail in an old paper of Kirillov and Neretin https://www.mat.univie.ac.at/~neretin/kiri87a.pdf  .

The case of $n=4$ which already shows many interesting phenomena (but already is nontrivial from the computer algebra viewpoint) is analysed in detail in the recent preprint https://arxiv.org/abs/2208.14631 - you might find it enlightening.

Linear independence does not really say much.

This algebraic variety is discussed in some detail in an old paper of Kirillov and Neretin: The variety $A_n$ of $n$-dimensional Lie algebra structures.

The case of $n=4$ which already shows many interesting phenomena (but already is nontrivial from the computer algebra viewpoint) is analysed in detail in the recent preprint Manivel, Sturmfels, and Sverrisdóttir - Four-Dimensional Lie Algebras Revisited — you might find it enlightening.