Let $\mathfrak{g}$ be a Lie algebra and let $[-,-]:\mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}$ be the Lie bracket. By a (Lie algebra) **deformation** of $\mathfrak{g}$ one usually means a (typically, analytic) one-parameter family of Lie brackets $[-,-]_t$ on the same underlying vector space $\mathfrak{g}$, agreeing with the original Lie bracket at $t=0$. A deformation is trivial if $[-,-]_t$ is obtained from $[-,-]$ by a one-parameter group $g(t)$ of linear transformations of $\mathfrak{g}$. The derivative at $t=0$ of $[-,-]_t$ defines a skew-symmetric bilinear map $\phi: \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}$ which is a cocycle in the Chevalley-Eilenberg complex $C^2(\mathfrak{g};\mathfrak{g})$, whereas if the deformation is trivial, this cocycle is actually the coboundary of some cochain in $C^1(\mathfrak{g};\mathfrak{g})$. Therefore, infinitesimal nontrivial deformations are classified by the Chevalley-Eilenberg cohomology group $H^2(\mathfrak{g};\mathfrak{g})$. You can work out easily what the cocycle condition by taking the derivative at $t=0$ of the Jacobi identity for the deformed bracket $[-,-]_t$.

You can work in a basis $X_a$ for $\mathfrak{g}$ and then $[-,-]_t$ is defined by structure "constants" (which are now a function of $t$!): $C_{ab}{}^c(t)$ and the cocycle is the derivative at $t=0$ of these structure constants.

Given a nontrivial infinitesimal deformation, one is typically interested in whether it will integrate to a one-parameter family of deformations. There is an infinite number of obstructions to integrability: each defined provided the previous one is overcome, which are classes in $H^3(\mathfrak{g};\mathfrak{g})$.

So, for example, if you can calculate $H^2(\mathfrak{g};\mathfrak{g})$ and $H^3(\mathfrak{g};\mathfrak{g})$ you are part of the way to understanding the deformation theory of $\mathfrak{g}$. Computing Chevalley-Eilenberg cohomology is simply a question of solving linear equations, so it is doable in principle and particularly amenable to computer calculations. But there are several results of a general nature which can save you lots of computation. For example, if $\mathfrak{g}$ is semisimple, then $H^2(\mathfrak{g};\mathfrak{g}) = 0$, so that such algebras are (infinitesimally) **rigid**. This means that any deformation is really just a $t$-dependent change of basis.

In your comment you mention rotational symmetry. The rotation algebra in dimension $d>2$ is semisimple, so in fact it is rigid. Any infinitesimal deformation you find is certainly trivial.

You can read about this in the original paper: C. Chevalley, S. Eilenberg, *Cohomology theory of Lie groups and Lie algebras*, Trans. Amer. Math. Soc. 63, (1948). 85–124. It is very well written and not very abstract at all. You may also benefit from the references in the answers to this MathOverflow question.