Schmidt representation of 3-qubits with real amplitudes

From the Schmidt representation we have that, up to local gates, every 2-qubit state can be written as $[\phi\rangle=\lambda_1 [00\rangle+\lambda_2 [11\rangle$with$\lambda_1$and$\lambda_2$ real numbers. For 3-qubits states, it is known, PRL 2000 85, that up to local gates every 3-qubit state can be written as $[\phi\rangle=\lambda_1 [000\rangle+\lambda_2 e^{i \theta }[100\rangle+\lambda_3[101\rangle+\lambda_4[110\rangle+\lambda_5[111\rangle$with$\lambda_i\ge0$and$0\le \theta \le \pi$. In this paper, we show that no every 3-qubit state with real amplitudes can be transform in the form $[\phi\rangle=\lambda_1 [000\rangle+\lambda_2 [100\rangle+\lambda_3[101\rangle+\lambda_4[110\rangle+\lambda_5[111\rangle$ by using local gates in the orthogonal group (the group generated by $R_y(\theta )$ and $X$ gates). We also show that, up to local gates in the orthogonal group, every 3-qubit with real amplitudes can be written as $[\phi\rangle=\lambda_1 [000\rangle+\lambda_2 [011\rangle+\lambda_3[101\rangle+\lambda_4[110\rangle+\lambda_5[111\rangle$withthe$\lambda_i$ real numbers. An explanation of this result can be found in the youtube video \url{https://youtu.be/gDN20QHzsoQ}