Explicit Hermitian scalar product on a simple complex Lie algebra

Every simple complex Lie algebra $$\mathfrak g$$ has a unique (up to isomorphism) compact real form $$\mathfrak k$$. That is, $$\mathfrak k$$ is a real Lie algebra with negative definite Killing form, and $$\mathfrak k\otimes_{\mathbb R}\mathbb C$$ is isomorphic to $$\mathfrak g$$.

If $$\mathfrak k$$ is the Lie algebra of a compact Lie group $$K$$, then the representation $$\mathfrak g$$ of $$K$$ has a $$K$$-invariant Hermitian scalar product given by $$(x,y)=\int_K\left\langle gx,gy\right\rangle dg$$ where $$\left\langle-,-\right\rangle$$ is the Killing form of $$\mathfrak g$$ and $$dg$$ is the Haar measure on $$K$$.

Question 0: are the above statements correct?

Question 1: is there a way to express the Hermitian scalar product $$(-,-)$$ on $$\mathfrak g$$ purely in Lie algebra terms, without referring to any Lie groups or Haar integration?

Your statements are not quite correct, since the Killing form on $$\mathfrak g$$ is complex bilinear rather than Hermitian, so what you produce is not a Hermitian form. But you can run the averaging-argument starting with any positive definite Hermitian form on $$\mathfrak g$$, there are no invariance properties required at this stage. However, in the algebraic picture, things can be obtained in a much simpler way.

Since $$\mathfrak k$$ is a real form of $$\mathfrak g$$, you can write any element of $$\mathfrak g$$ as $$X+iY$$ for $$X,Y\in\mathfrak k$$. Now you can simply use the negative of the real Killing form $$B$$ on $$\mathfrak k$$ (which is negative definite since $$\mathfrak k$$ is compact and semisimple and $$K$$-invariant) and extend it to a Hermitian form on $$\mathfrak g$$. This means that you define $$\langle X_1+iY_1,X_2+iY_2\rangle:=B(X_1,X_2)+B(Y_1,Y_2)+i(B(X_2,Y_1)-B(X_1,Y_2))$$.

The standard description for this is via a Cartan-involution $$\theta$$ on $$\mathfrak g$$, which by definition has the property that evaluating the negative of the Killing form of $$\mathfrak g$$ on $$X$$ and $$\theta(Y)$$ defines a positive definite inner product. Then $$\mathfrak k$$ is obtained as the fixed point set of $$\theta$$.