Given a Lie algebra $\mathfrak{g}$, we can define its Killing form $$K(x,y) = \mathrm{Tr}(ad_x\circ ad_y)$$for $x, y\in \mathfrak g$.
Whilst I understand that the Cartan decomposition $$\mathfrak g =\mathfrak h \oplus\left(\bigoplus_{\alpha\in R^+}(\mathfrak g_\alpha \oplus\mathfrak g_{-\alpha})\right)$$is orthogonal with respect to the Killing form, I'm still not really sure how to visualise the Killing form - by which I mean, intuitively, given $x, y\in \mathfrak g$, is there a good way to think about what the Killing form should be without pure calculation.
I'm also unsure what the reason for the definition is - what is the significance of the Trace and the adjoint action?
I would like to understand:
How, given $x, y\in \mathfrak g$, can I visualise the Killing form?
What is the reason behind the definition of the Killing form?