This question comes from analysis the differential geometry.
What proof thought is? I don't know it.
The 1st question: how to proof the 4 identical relations.
The 2nd question: why ${R_{1212}}$ is only independent variant?
$$ \begin{align} R_{mijk} = \sum_\ell g_{m\ell} R_{ijk}^\ell , \quad m,i,j,k = 1,2.\\[6pt] \begin{cases} R_{mijk} = - R_{imjk},\\ R_{mijk} = - R_{mikj},\\ R_{mijk} = R_{jkmi}, \\ R_{mijk} + R_{mjki} + R_{mkij} = 0. \end{cases} \end{align} $$
I only give some steps, but I didn't proof them. Because of difficulty.
Steps1:
$$ \begin{array}{l} {R_{mijk}} = \sum\limits_l {{g_{ml}}R_{ijk}^l} = {g_{m1}}R_{ijk}^1 + {g_{m2}}R_{ijk}^2 = {g_{m1}}\left( {\frac{{\partial \Gamma _{ij}^1}}{{\partial {u^k}}} - \frac{{\partial \Gamma _{ik}^1}}{{\partial {u^j}}} + \sum\limits_p {\left( {\Gamma _{ij}^p\Gamma _{pk}^1 - \Gamma _{ik}^p\Gamma _{pj}^1} \right)} } \right)\\ + {g_{m2}}\left( {\frac{{\partial \Gamma _{ij}^2}}{{\partial {u^k}}} - \frac{{\partial \Gamma _{ik}^2}}{{\partial {u^j}}} + \sum\limits_p {\left( {\Gamma _{ij}^p\Gamma _{pk}^2 - \Gamma _{ik}^p\Gamma _{pj}^2} \right)} } \right)\\ {R_{imjk}} = \sum\limits_l {{g_{il}}R_{mjk}^l} = {g_{i1}}R_{mjk}^1 + {g_{i2}}R_{mjk}^2 = {g_{i1}}\left( {\frac{{\partial \Gamma _{mj}^1}}{{\partial {u^k}}} - \frac{{\partial \Gamma _{mk}^1}}{{\partial {u^j}}} + \sum\limits_p {\left( {\Gamma _{mj}^p\Gamma _{pk}^1 - \Gamma _{mk}^p\Gamma _{pj}^1} \right)} } \right)\\ + {g_{i2}}\left( {\frac{{\partial \Gamma _{mj}^2}}{{\partial {u^k}}} - \frac{{\partial \Gamma _{mk}^2}}{{\partial {u^j}}} + \sum\limits_p {\left( {\Gamma _{mj}^p\Gamma _{pk}^2 - \Gamma _{mk}^p\Gamma _{pj}^2} \right)} } \right) \end{array} $$
Steps2:
$$ \begin{array}{l} \sum\limits_p {\left( {{g_{i1}}\Gamma _{mj}^p\Gamma _{pk}^1 - {g_{i1}}\Gamma _{mk}^p\Gamma _{pj}^1} \right)} \\ \sum\limits_p {\left( {{g_{m1}}\Gamma _{ij}^p\Gamma _{pk}^1 - {g_{m1}}\Gamma _{ik}^p\Gamma _{pj}^1} \right)} \\ {g_{i1}}\Gamma _{mj}^p = ? - {g_{m1}}\Gamma _{ij}^p\\ \Gamma _{mj}^p = \sum\limits_l {{g^{pl}}} [mj,l] = \sum\limits_l {{g^{pl}}} {r_{mj}} \cdot {r_l}\\ \Gamma _{ij}^p = \sum\limits_l {{g^{pl}}} [ij,l] = \sum\limits_l {{g^{pl}}} {r_{ij}} \cdot {r_l}\\ {g_{i1}}\Gamma _{mj}^p = \sum\limits_l {{g^{pl}}} {g_{i1}} \cdot {r_{mj}} \cdot {r_l} = \sum\limits_l {{g^{pl}}} {r_i} \cdot {r_1} \cdot {r_{mj}} \cdot {r_l}\\ {g_{m1}}\Gamma _{ij}^p = \sum\limits_l {{g^{pl}}} {g_{m1}} \cdot {r_{ij}} \cdot {r_l} = \sum\limits_l {{g^{pl}}} {r_m} \cdot {r_1} \cdot {r_{ij}} \cdot {r_l}\\ {r_i} \cdot {r_{mj}} = ? - {r_m} \cdot {r_{ij}} \end{array} $$
Steps3:
$$ \begin{array}{l} {r_i} \cdot {r_{mj}} = ? - {r_m} \cdot {r_{ij}}\\ {r_{mj}} \cdot {r_i} = \frac{1}{2}\left( {\frac{{\partial {g_{mi}}}}{{\partial {u^j}}} + \frac{{\partial {g_{ji}}}}{{\partial {u^m}}} - \frac{{\partial {g_{mj}}}}{{\partial {u^i}}}} \right)\\ {r_{ij}} \cdot {r_m} = \frac{1}{2}\left( {\frac{{\partial {g_{im}}}}{{\partial {u^j}}} + \frac{{\partial {g_{jm}}}}{{\partial {u^i}}} - \frac{{\partial {g_{ij}}}}{{\partial {u^m}}}} \right)\\ {r_i} \cdot {r_{mj}} \ne - {r_m} \cdot {r_{ij}}??? \end{array} $$
PS: The definition you need is here:
$$ \left\{ {\begin{array}{*{20}{c}} {{r_{ij}} \cdot {r_l} = \frac{1}{2}\left( {\frac{{\partial {g_{il}}}}{{\partial {u^j}}} + \frac{{\partial {g_{jl}}}}{{\partial {u^i}}} - \frac{{\partial {g_{ij}}}}{{\partial {u^l}}}} \right) = [ij,l] = \sum\limits_k {\Gamma _{ij}^k{g_{kl}}} ,k = 1,2.}\\ {\Gamma _{ij}^k = \sum\limits_l {{g^{kl}}} [ij,l]\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}\begin{array}{*{20}{c}} {} \end{array}} \end{array}} \right. $$
$$ R_{ijk}^l = \frac{{\partial \Gamma _{ij}^l}}{{\partial {u^k}}} - \frac{{\partial \Gamma _{ik}^l}}{{\partial {u^j}}} + \sum\limits_p {\left( {\Gamma _{ij}^p\Gamma _{pk}^l - \Gamma _{ik}^p\Gamma _{pj}^l} \right)} ,i,j,k,l,p = 1,2. $$
Anyone can help me?