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I'm working with MAPLE. I have a Lie algebra spanned by $\{e_1,\ldots,e_8\}$ and the Levi-Civita connection associated to the Riemannian metric corresponding to the inner product satisfying that $\{e_i\}$ is orthonormal.

I have already computed (as matrices) $\nabla_{e_i}$ for $i\in\{1,\ldots,8\}$ using Koszul Formula . Here each $\nabla_{e_i}$ is an $8\times 8$ matrix.

Now I want to compute the curvature tensors $R(e_i,e_j)$, $i,j\in \{1..8\}$. This can be done using $R(e_i,e_j)=\nabla_{e_i} \nabla_{e_j} -\nabla_{e_j} \nabla_{e_i}-\nabla_{[e_i,e_j]}$.

My problem is to compute the last term. I've thought of defining a linear function of the $e_i$'s in terms of the matrices $\nabla_{e_i} $ but I don't know how to do it. I was reading the Maple manual but I couldn't find anything.

Thank you in advance

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  • $\begingroup$ Curvature is determined by the structure constants of the Lie algebra. Look at standard Riemannian geometry and Lie group texts like Helgason, for example. $\endgroup$ Commented Feb 17 at 1:11

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