using $exp(\xi \cdot x^2)$
Try again with two vectors?
$g_{ab} = e_{ab} exp(\xi_{uv} x^u x^v)$ for symmetric $\xi_{uv}$
$g_{ab,c} = e_{ab} exp(\xi_{uv} x^u x^v) (\xi_{uv} x^u x^v)_{,c}$
$g_{ab,c} = g_{ab} \cdot 2 \xi_{cu} x^u$
$\Gamma_{abc} = (g_{ab} \xi_{cu} + g_{ac} \xi_{bu} - g_{bc} \xi_{au}) x^u$
${\Gamma^a}_{bc} = (\delta^a_b \xi_{cu} + \delta^a_c \xi_{bu} - g_{bc} g^{ae} \xi_{eu}) x^u$
${\Gamma^a}_{bc,d} =
(\delta^a_b \xi_{cu} + \delta^a_c \xi_{bu} - g_{bc} g^{ae} \xi_{eu}) \delta^u_d
+ (\delta^a_b \xi_{cu} + \delta^a_c \xi_{bu} - g_{bc} g^{ae} \xi_{eu})_{,d} x^u
$
$= \delta^a_b \xi_{cd} + \delta^a_c \xi_{bd} - g_{bc} g^{ae} \xi_{ed} - (g_{bc} g^{ae})_{,d} \xi_{eu} x^u$
$= \delta^a_b \xi_{cd} + \delta^a_c \xi_{bd} - g_{bc} g^{ae} \xi_{ed} - (g_{bc,d} g^{ae} + g_{bc} g^{af} g_{fg,d} g^{ge}) \xi_{eu} x^u$
$= \delta^a_b \xi_{cd} + \delta^a_c \xi_{bd} - g_{bc} g^{ae} \xi_{ed} - 2 (g_{bc} g^{ae} \xi_{dv} x^v - g_{bc} g^{af} g_{fg} g^{ge} \xi_{dv} x^v) \xi_{eu} x^u$
$= \delta^a_b \xi_{cd} + \delta^a_c \xi_{bd} - g_{bc} g^{ae} \xi_{ed} - 2 (g_{bc} g^{ae} - g_{bc} g^{ae}) \xi_{dv} x^v \xi_{eu} x^u$
$= \delta^a_b \xi_{cd} + \delta^a_c \xi_{bd} - g_{bc} g^{ae} \xi_{ed}$
${\Gamma^a}_{uc} {\Gamma^u}_{bd} =
(\delta^a_u \xi_{cv} + \delta^a_c \xi_{uv} - g_{uc} g^{ae} \xi_{ev}) x^v
(\delta^u_b \xi_{dw} + \delta^u_d \xi_{bw} - g_{bd} g^{uf} \xi_{fw}) x^w
$
$ = (
+ \delta^a_u \xi_{cv} \delta^u_b \xi_{dw}
+ \delta^a_u \xi_{cv} \delta^u_d \xi_{bw}
- \delta^a_u \xi_{cv} g_{bd} g^{ue} \xi_{ew}
+ \delta^a_c \xi_{uv} \delta^u_b \xi_{dw}
+ \delta^a_c \xi_{uv} \delta^u_d \xi_{bw}
- \delta^a_c \xi_{uv} g_{bd} g^{ue} \xi_{ew}
- g_{uc} g^{ae} \xi_{ev} \delta^u_b \xi_{dw}
- g_{uc} g^{ae} \xi_{ev} \delta^u_d \xi_{bw}
+ g_{uc} g^{ae} \xi_{ev} g_{bd} g^{uf} \xi_{fw}
) x^v x^w$
$ = (
+ \delta^a_b \xi_{cv} \xi_{dw}
+ \delta^a_d \xi_{cv} \xi_{bw}
- g_{bd} g^{ae} \xi_{cv} \xi_{ew}
+ \delta^a_c \xi_{bv} \xi_{dw}
+ \delta^a_c \xi_{dv} \xi_{bw}
- \delta^a_c g_{bd} g^{ue} \xi_{uv} \xi_{ew}
- g_{cd} g^{ae} \xi_{ev} \xi_{bw}
+ g_{bd} g^{ae} \xi_{ev} \xi_{cw}
- g_{bc} g^{ae} \xi_{ev} \xi_{dw}
) x^v x^w$
${R^a}_{bcd} =
+ \delta^a_b \xi_{cd}
+ \delta^a_d \xi_{bc}
- g_{bd} g^{ae} \xi_{ce}
- \delta^a_b \xi_{cd}
- \delta^a_c \xi_{bd}
+ g_{bc} g^{ae} \xi_{ed}
+ (
+ \delta^a_b \xi_{cv} \xi_{dw}
+ \delta^a_d \xi_{cv} \xi_{bw}
- g_{bd} g^{ae} \xi_{cv} \xi_{ew}
+ \delta^a_c \xi_{bv} \xi_{dw}
+ \delta^a_c \xi_{dv} \xi_{bw}
- \delta^a_c g_{bd} g^{ue} \xi_{uv} \xi_{ew}
- g_{cd} g^{ae} \xi_{ev} \xi_{bw}
+ g_{bd} g^{ae} \xi_{ev} \xi_{cw}
- g_{bc} g^{ae} \xi_{ev} \xi_{dw}
) x^v x^w
- (
+ \delta^a_b \xi_{dv} \xi_{cw}
+ \delta^a_c \xi_{dv} \xi_{bw}
- g_{bc} g^{ae} \xi_{dv} \xi_{ew}
+ \delta^a_d \xi_{bv} \xi_{cw}
+ \delta^a_d \xi_{cv} \xi_{bw}
- \delta^a_d g_{bc} g^{ue} \xi_{uv} \xi_{ew}
- g_{cd} g^{ae} \xi_{ev} \xi_{bw}
+ g_{bc} g^{ae} \xi_{ev} \xi_{dw}
- g_{bd} g^{ae} \xi_{ev} \xi_{cw}
) x^v x^w
$
$=
- \delta^a_c \xi_{bd}
+ \delta^a_d \xi_{bc}
+ g_{bc} g^{ae} \xi_{de}
- g_{bd} g^{ae} \xi_{ce}
+ (
\delta^a_c \xi_{bv} \xi_{dw}
- \delta^a_c g_{bd} g^{ue} \xi_{uv} \xi_{ew}
- \delta^a_d \xi_{cv} \xi_{bw}
+ \delta^a_d g_{bc} g^{ue} \xi_{uv} \xi_{ew}
- g_{bc} g^{ae} \xi_{dv} \xi_{ew}
+ g_{bd} g^{ae} \xi_{cv} \xi_{ew}
) x^v x^w
$
$R_{ab} =
- \delta^c_c \xi_{ab}
+ \delta^c_b \xi_{ac}
+ g_{ac} g^{ce} \xi_{be}
- g_{ab} g^{ce} \xi_{ce}
+ (
\delta^c_c \xi_{av} \xi_{bw}
- \delta^c_c g_{ab} g^{ue} \xi_{uv} \xi_{ew}
- \delta^c_b \xi_{cv} \xi_{aw}
+ \delta^c_b g_{ac} g^{ue} \xi_{uv} \xi_{ew}
- g_{ac} g^{ce} \xi_{bv} \xi_{ew}
+ g_{ab} g^{ce} \xi_{cv} \xi_{ew}
) x^v x^w
$
$=
- 2 \xi_{ab}
- g_{ab} g^{cd} \xi_{cd}
+ 2 (
\xi_{av} \xi_{bw}
- g_{ab} g^{cd} \xi_{cv} \xi_{dw}
) x^v x^w
$
$R = g^{ab} (
- 2 \xi_{ab}
- g_{ab} g^{cd} \xi_{cd}
+ 2 (
\xi_{av} \xi_{bw}
- g_{ab} g^{cd} \xi_{cv} \xi_{dw}
) x^v x^w
)$
$ = - 6 g^{ab} (\xi_{ab} + \xi_{av} \xi_{bw} x^v x^w)$
This only really works if $R$ is a function of $x^a x^b$...
Unless $\xi_{au} (e^{-1})^{ub} = 0$
Then we get $R = 0$
Substitute...
$R_{ab} = 2 (-\xi_{ab} + \xi_{av} \xi_{bw} x^v x^w)$
So this only works for $R_{ab} = 0$