To solve the expression \( B = \left(2 + \frac{1}{3150}\right) \cdot \frac{3}{6517} - \frac{1}{1050} \cdot \left(3 + \frac{6561}{6517}\right) + \frac{4}{1050} - \frac{6}{3150} \cdot 6517 \), let's break it down step by step.

1. Simplify the terms inside the parentheses:
\[
2 + \frac{1}{3150} = 2 + 0.00031746 \approx 2.00031746
\]
\[
3 + \frac{6561}{6517} = 3 + 1.00675 \approx 4.00675
\]

2. Calculate the products and quotients:
\[
\left(2.00031746\right) \cdot \frac{3}{6517} = 2.00031746 \cdot 0.0004602 \approx 0.0009206
\]
\[
\frac{1}{1050} \cdot 4.00675 = 0.00095238 \cdot 4.00675 \approx 0.003812
\]
\[
\frac{4}{1050} = 0.00380952
\]
\[
\frac{6}{3150} \cdot 6517 = 0.00190476 \cdot 6517 \approx 12.4
\]

3. Combine all the terms:
\[
B = 0.0009206 - 0.003812 + 0.00380952 - 12.4
\]

4. Simplify the final expression:
\[
B \approx 0.0009206 - 0.003812 + 0.00380952 - 12.4
\]
\[
B \approx 0.0009206 - 0.003812 + 0.00380952 - 12.4
\]
\[
B \approx 0.0009206 - 0.00000248 - 12.4
\]
\[
B \approx 0.00091812 - 12.4
\]
\[
B \approx -12.39908188
\]

So, the value of \( B \) is approximately \(-12.39908188\).