A4,1

ShowSol[A01, ( {{0, b_2, b_3, b_4}, {b_2, b_5, b_6, b_7}, {b_3, b_6, b_8, b_9}, {b_4, b_7, b_9, b_10}} ), {b_2→0, b_3→ -b_5}]

We consider the Lie-algebra with commutator<br /> ad = ( {{0, 0, 0, 0}, {0, 0, 0, e_1}, {0, 0, 0, e_2}, {0, -e_1, -e_2, 0}} )

and the scalar product of the form ℬ = ( {{0, 0, -b_5, b_4}, {0, b_5, b_6, b_7}, {-b_5, b_6, b_8, b_9}, {b_4, b_7, b_9, b_10}} )

with determinant |ℬ| = b_4^2 b_6^2 + 2 b_4 b_5 b_6 b_7 + b_5^2 b_7^2 - b_4^2 b_5 b_8 - 2 b_4 b_5^2 b_9 - b_5^3 b_10 .

Then, the non-zero evaluations of the Riemannian curvature tensor are determined by<br />ℛ-Flat

The Ricci curvature tensor is zero…4×4

ShowSol[A01, ( {{0, b_2, b_3, b_4}, {b_2, b_5, b_6, b_7}, {b_3, b_6, b_8, b_9}, {b_4, b_7, b_9, b_10}} ), {b_2→0, b_3→b_5}]

We consider the Lie-algebra with commutator<br /> ad = ( {{0, 0, 0, 0}, {0, 0, 0, e_1}, {0, 0, 0, e_2}, {0, -e_1, -e_2, 0}} )

and the scalar product of the form ℬ = ( {{0, 0, b_5, b_4}, {0, b_5, b_6, b_7}, {b_5, b_6, b_8, b_9}, {b_4, b_7, b_9, b_10}} )

with determinant |ℬ| = b_4^2 b_6^2 - 2 b_4 b_5 b_6 b_7 + b_5^2 b_7^2 - b_4^2 b_5 b_8 + 2 b_4 b_5^2 b_9 - b_5^3 b_10 .

The Ricci curvature tensor is zero…4×4

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