Dear MathOverFlow: As many of you know, the Lie algebra of the group of $3 \times 3$ orthogonal matrices (with determinant one) is isomorphic to $\Bbb{R}^3$ endowed with the cross-product $a \times b$; furthermore the Killing form in this case is the usual dot-product $a \cdot b$. Linear algebra teaches us that these two operations on $\Bbb{R}^3$ are related by an elegant $2 \times 2$ determinantal identity --- a special case of what is known as a {\it contraction formula} --- namely:

$$ \big( a \times b \big) \cdot \big( c \times d \big) \ = \ \text{det} \, \begin{pmatrix} a \cdot c & a \cdot d \\ b \cdot c & b \cdot d \end{pmatrix} $$

My question is: If $\frak{g}$ is the Lie Algebra (say of a compact group), is there some 'contraction identity' for

$$ \big[ a , b \big] \, \cdot \, \big[ c , d \big] $$

where $a,b,c,d \in \frak{g}$ and where $\cdot$ denotes the Killing form ?

Kind regards for the consideration of this question.