= Solution
For the <Lie algebra> $\mathfrak g$ of a <Lie group> $G$, let $\operatorname{ad}_X(Y)=[X,Y]$. The <Killing form> is the symmetric <bilinear form>
$$
B(X,Y)=\operatorname{tr}(\operatorname{ad}_X\operatorname{ad}_Y).
$$
For a finite-dimensional real <Lie algebra>, the <Cartan criterion for semisimplicity> says that $B$ is nondegenerate precisely when $\mathfrak g$ is a <semisimple Lie algebra>. Thus a <semisimple Lie group> supplies the required condition. Compactness alone is insufficient: an abelian <Lie algebra>, including that of a torus, has zero <Killing form>.
The <Jacobi identity> gives $\operatorname{ad}_{[Z,X]}=[\operatorname{ad}_Z,\operatorname{ad}_X]$. Cyclic invariance of the <trace> therefore gives
$$
B([Z,X],Y)+B(X,[Z,Y])=0.
$$
There is also invariance under the full <Adjoint representation of a Lie group>: $\operatorname{ad}_{\operatorname{Ad}_aX}=\operatorname{Ad}_a\operatorname{ad}_X\operatorname{Ad}_{a^{-1}}$, so $B(\operatorname{Ad}_aX,\operatorname{Ad}_aY)=B(X,Y)$.
Define a <pseudo-Riemannian metric> by translating $B$ from the identity:
$$
g_a\big((L_a)_*X,(L_a)_*Y\big)=B(X,Y).
$$
It is smooth and nondegenerate. Left translations preserve it by construction. Right translation, expressed in left-translated tangent coordinates, acts by $\operatorname{Ad}_{b^{-1}}$, so it also preserves the metric. This constructs a <bi-invariant pseudo-Riemannian metric>.
For <left-invariant vector fields>, the derivative terms in the <Koszul formula> vanish. Invariance of $B$ then gives
$$
2g(\nabla_XY,Z)=g([X,Y],Z)-g([Y,Z],X)+g([Z,X],Y)=g([X,Y],Z),
\qquad \nabla_XY=\tfrac12[X,Y].
$$
Use the <Riemann curvature tensor> convention $R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$. Substituting the <Levi-Civita connection> and using the <Jacobi identity> yields
$$
R(X,Y)Z=\tfrac14\big([X,[Y,Z]]-[Y,[X,Z]]\big)-\tfrac12[[X,Y],Z]
=-\tfrac14[[X,Y],Z].
$$
The <Ricci tensor> is the trace of $X\mapsto R(X,Y)Z$. Since $[[X,Y],Z]=\operatorname{ad}_Z\operatorname{ad}_Y X$, it follows that
$$
\boxed{\operatorname{Ric}(Y,Z)=-\tfrac14B(Y,Z)=-\tfrac14g(Y,Z).}
$$
This is the <Killing-form Einstein metric>. Rescaling to $g=cB$, with nonzero constant $c$, gives <Einstein metric> constant $-1/(4c)$.
For a compact <semisimple Lie group>, $B$ is negative definite and $-B$ is a positive-definite <bi-invariant Riemannian metric>. For example, $SU(2)$ has topology $S^3$ and <metric signature> $(0,3)$ for $B$, while $SO(3)$ has the same <Lie algebra> and <metric signature>, but topology $\mathbb{RP}^3$. The <special linear group> $SL(2,\mathbb R)$ has <metric signature> $(2,1)$ and topology $S^1\times\mathbb R^2$; its <universal covering Lie group> has topology $\mathbb R^3$ and the same <Killing form>. The real <Lie algebra> of $SL(2,\mathbb C)$ has <metric signature> $(3,3)$, and the group has topology $S^3\times\mathbb R^3$. Thus the <Killing form> distinguishes compact from noncompact directions in these semisimple examples, but does not determine global topology or distinguish different covering groups.
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