# Definition Given a [[matrix Lie Group]] $G$ and its Lie algebra $\mathfrak{g}$, $\mathfrak{g}$ comes equipped with a symmetric $(2,0)$ [[tensor]] known as the *Killing form*, defined by: $ K(X,Y) \equiv - \text{tr}(\text{ad}_X \text{ad}_Y) $ where $X, Y \in \mathfrak{g}$ and $\text{ad}_\cdot$ is the [[Ad Homomorphism 1]]. $\text{tr}$ is the [[trace]]. $\text{ad}_X \text{ad}_Y$ is an abuse of notation and means $\text{ad}_X \circ \text{ad}_Y$ # Properties Using the [[Ad Homomorphism 1]], and given $A \in G$ and $X, Y \in \mathfrak{g}$, we have the property: $ \text{ad}_{\text{Ad}_A(X)} = \text{Ad}_{A} \circ \text{ad}_X \circ \text{Ad}_{A^{-1}} $ $ \begin{align} K(\text{Ad}_A(X), \text{Ad}_A(Y)) &= -\text{tr}(\text{ad}_{\text{Ad}_A(X)}\text{ad}_{\text{Ad}_A(Y)})\\ &= -\text{tr}(\text{Ad}_{A} \circ \text{ad}_X \circ \text{Ad}_{A^{-1}} \circ \text{Ad}_{A} \circ \text{ad}_Y \circ \text{Ad}_{A^{-1}})\\ &= -\text{tr}(\text{Ad}_{A} \circ \text{ad}_X \circ (\text{Ad}_A)^{-1} \circ \text{Ad}_{A} \circ \text{ad}_Y \circ \text{Ad}_{A^{-1}})\\ &= -\text{tr}(\text{Ad}_{A} \circ \text{ad}_X \circ \text{ad}_Y \circ \text{Ad}_{A^{-1}})\\ &= -\text{tr}((\text{Ad}_A)^{-1} \circ \text{Ad}_{A} \circ \text{ad}_X \circ \text{ad}_Y\\ &= -\text{tr}(\text{ad}_X \circ \text{ad}_Y)\\ &= K(X,Y)\\ \end{align} $ where we have used the property of [[Group Homomorphism|homomorphisms]] to get $\text{Ad}_A^{-1} = (\text{Ad}_A)^{-1}$ and the cyclic property of the trace to rearrange terms.