Let $A^{\alpha}_\mu$ be the gauge field of a Yang-Mills theory where $\alpha$ is the gauge index of generators for some Lie algebra with structure constant $C_{\alpha \beta}^\gamma$ and $\mu$ is the space-time index.

According to Wiki article, the field strength tensor is given by \begin{equation} F^\alpha_{\mu \nu}:=\partial_\mu A^\alpha_\nu - \partial_\nu A^\alpha_\mu + C_{\alpha \beta}^\gamma A^\alpha_\mu A^\beta_\nu \end{equation} and the Lagrangian is \begin{equation} L_{YM} := F^{\alpha \mu \nu} F^\alpha_{\mu \nu} \text{ up to an overall normalization factor } \end{equation}

Now, expanding every term in the above $L_{YM}$, we roughly have \begin{equation} L_{YM} \simeq (\partial_\mu A_\nu^\alpha)^2 - \Bigl[C_{\alpha \beta}^\gamma A^\alpha_\mu A^\beta_\nu\partial_\mu A^\gamma_\nu \Big] + \Bigl[ C_{\alpha \beta}^\gamma C_{\alpha' \beta'}^\gamma A^\alpha_\mu A^\beta_\nu A^{\alpha' \mu} A^{\beta' \nu} \Bigr] \end{equation} where I have omitted numerical factors in each term.

Obviously the first term $(\partial_\mu A_\nu^\alpha)^2$ can be interpreted as the "free theory part". On the other hands, the two terms in $[ \cdots]$ are "self-interacting parts" Moreover, the third term in $L_{YM}$ above looks like a "quartic" interaction.

Now my question is that

What sort of interaction does the second term $-C_{\alpha \beta}^\gamma A^\alpha_\mu A^\beta_\nu\partial_\mu A^\gamma_\nu$ stand for? Is it some sort of "cubic" interaction because there are only three factors of $A$? However, presence of the derivative $\partial_\mu$ seems to complicate things...

Could anyone please clarify for me?