Note first the following definition of the Riemann curvature tensor I have been using:
$$\text{Riem}(\omega, Z, X, Y) := \omega\ (\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z - \nabla_{[X,Y]}Z)$$ where the components can be written in terms of the connection coefficients as
$$\text{Riem}^{i}_{jkm} = \partial_{k}\Gamma^{i}_{jm}-\partial_{m}\Gamma^{i}_{jk} + \Gamma^{r}_{jm}\Gamma^{i}_{rk} - \Gamma^{t}_{jk}\Gamma^{i}_{tm}.$$
Consider the round sphere $(S^{2},\mathcal{O}, \mathcal{A})$ and a chart $(U,x) \in \mathcal{A}$ where the only non-vanishing connection coefficients are given by $$\Gamma^{1}_{22}(x^{-1}(\theta, \phi)) = -\sin\theta\cos\theta \ \ \text{and} \ \ \Gamma^{2}_{12}(x^{-1}(\theta, \phi))=\Gamma^{2}_{21}(x^{-1}(\theta, \phi))=\cot\theta$$ for $\phi \in (0, 2\pi)$ and $\theta \in (0, \pi)$. We need to calculate the component functions $\text{Riem}^{1}_{212}$ and $\text{Riem}^{1}_{112}$ and give a list of the remaining component functions.
My question is not about the first part, calculating $\text{Riem}^{1}_{212}$ and $\text{Riem}^{1}_{112}$, which are found to be
$$\text{Riem}^{1}_{212} = \sin^{2}\theta, \ \ \ \ \text{Riem}^{1}_{112}=0$$ but the second part: "...give a list of the remaining component functions". The lecturer in this video at minute 32:30 says the only remaining component functions can be found as $\text{Riem}^{1}_{212} = -\text{Riem}^{1}_{221}$, $\text{Riem}^{1}_{112} = -\text{Riem}^{1}_{121}$, $\text{Riem}^{2}_{212} = -\text{Riem}^{2}_{221}$, and $\text{Riem}^{2}_{112} = - \text{Riem}^{2}_{121}$, but in this example aren't there $16$ component functions in total? I know also that since $\text{dim}\mathcal{S}^{2} = 2$ there are $4$ independent components, but I'm not seeing why he said these are the only remaining ones. Maybe the question should be phrased "Give a list of the remaining component functions that correspond to the non-vanishing gammas provided"?
Thank you for your time.