Sectional curvature to Ricci tensor

We know that to measure the if the n-dimensional manifold is curved or not, we look at the Riemann Tensor. If it is 0, then it is flat otherwise it is curved positively or negatively (which each of the sign has its own geometric meaning).

Ricci tensor help us track the volume changes along when we move in the direction of a geodesic.

We know the well known method of contracting the indices gives us the direct result from Riemann tensor to Ricci tensor. But this doesn’t gives a meaning of what does it means geometrically. So we us Sectional Curvature.

Sectional curvature (K) helps us understand geometrically that whether the geodesics are converging, diverging or remains flat when we move along a direction of vector ‘v’. ‘

So Ricci Tensors are derived by selecting direction ‘v’ and summing over all possible Sectional Curvatures containing the vector ‘v’.

I am having a problem deriving the Ricci tensor from Sectional curvature tensor. Any help would be grateful.

K = $v^j$$v^k$$[R(e_i,e_j)]e_k$.$e_i$ gives

$v^j$$v^k$$R^l$$_k$$_i$$_j$ $e_l$.$e_i$

Then

$v^j$$v^k$$R^l$$_k$$_i$$_j$ $g$$_l$$_i$

Finally,

$v^j$$v^k$$R$$_i$$_k$$_i$$_j$

Which upon summation on i gives

Ricci Tensor (Ric (v,v))= $v^j$$v^k$$R$$_k$$_j$

Which is a component of a ricci tensor..

here we can see that the lower indices of the Riemann Tensor are contracted which is wrong. The upper index and the lower middle index are contracted which is correct.

Please tell me the problem here.