  The structure tensor of $\mathfrak{sl}_n$, denoted $T_{\mathfrak{sl}_n}$, is
the tensor arising from the Lie bracket bilinear operation on the set of
traceless $n \times n$ matrices over $\mathbb{C}$. This tensor is intimately
related to the well studied matrix multiplication tensor. Studying the
structure tensor of $\mathfrak{sl}_n$ may provide further insight into the
complexity of matrix multiplication and the "hay in a haystack" problem of
finding explicit sequences tensors with high rank or border rank. We aim to
find new bounds on the rank and border rank of this structure tensor in the
case of $\mathfrak{sl}_3$ and $\mathfrak{sl}_4$. We additionally provide bounds
in the case of the lie algebras $\mathfrak{so}_4$ and $\mathfrak{so}_5$. The
lower bounds on the border ranks were obtained via various recent techniques,
namely Koszul flattenings, border substitution, and border apolarity. Upper
bounds on the rank of $T_{\mathfrak{sl}_3}$ are obtained via numerical methods
that allowed us to find an explicit rank decomposition.
