Constructing a $3 \times 3$ Principal curvature tensor (sometimes just called a curvature tensor) in the global coordinate system

Question

I have seen several sources ("Consistent Computation of First and Second-Order Differential Quantities for Surface Meshes" as an example) using the equation $$\mathbf{C}=\kappa_1\mathbf{\hat{e}_1\hat{e}_1^T} + \kappa_2\mathbf{\hat{e}_2\hat{e}_2^T}$$ in a local coordinate system for the $(3\times3)$ principal curvature tensor $\mathbf{C}$ (I understand its not accurate to call it the curvature tensor, but that's what most people refer to it as). Here $\kappa_1$ and $\kappa_2$ are the principal curvatures and $\mathbf{\hat{e}_1}$ and $\mathbf{\hat{e}_2}$ are the principal directions in the local coordinate system. We can then transform the "Curvature Tensor" to the global coordinate system using $$\mathbf{C_g}=\mathbf{QCQ^T}$$ (Here $\mathbf{Q}$ is an orthogonal column matrix comprised of the 3 coordinate axes of the local coordinate system in term of the global coordinate system. So essentially $2$ tangential vectors and $1$ normal vector to a given surface at a point, but all in terms of global coordinates). Could we instead construct the $3 \times 3$ principal curvature tensor directly from the principal directions, if we already had them in terms of the global coordinate system? So I would use the same equation for $\mathbf{C_g}$ as for $\mathbf{C}$ but with principal directions $\mathbf{\hat{e}_1}$ and $\mathbf{\hat{e}_2}$ now in terms of the global coordinate system. That would essentially mean doing $$\mathbf{C_g}=\kappa_1\mathbf{Q\hat{e}_1\hat{e}_1^TQ^T}+\kappa_2\mathbf{Q\hat{e}_2\hat{e}_2^TQ^T}$$ where I would be performing the transformation on the tensors $\mathbf{\hat{e}_1\hat{e}_1^T}$ and $\mathbf{\hat{e}_2\hat{e}_2^T}$ instead of on $\mathbf{C}$. Of course I wouldn't actually be doing this transformation since I already have the principal directions in the global coordinate system. (I am not sure, but I guess this question is more about rules of tensor transformation than Differential Geometry)